diff --git a/.gitignore b/.gitignore index b7faf40..4e3bf1e 100644 --- a/.gitignore +++ b/.gitignore @@ -205,3 +205,87 @@ cython_debug/ marimo/_static/ marimo/_lsp/ __marimo__/ + +# Covers JetBrains IDEs: IntelliJ, GoLand, RubyMine, PhpStorm, AppCode, PyCharm, CLion, Android Studio, WebStorm and Rider +# Reference: https://intellij-support.jetbrains.com/hc/en-us/articles/206544839 + +# User-specific stuff +.idea/**/workspace.xml +.idea/**/tasks.xml +.idea/**/usage.statistics.xml +.idea/**/dictionaries +.idea/**/shelf + +# AWS User-specific +.idea/**/aws.xml + +# Generated files +.idea/**/contentModel.xml + +# Sensitive or high-churn files +.idea/**/dataSources/ +.idea/**/dataSources.ids +.idea/**/dataSources.local.xml +.idea/**/sqlDataSources.xml +.idea/**/dynamic.xml +.idea/**/uiDesigner.xml +.idea/**/dbnavigator.xml + +# Gradle +.idea/**/gradle.xml +.idea/**/libraries + +# Gradle and Maven with auto-import +# When using Gradle or Maven with auto-import, you should exclude module files, +# since they will be recreated, and may cause churn. Uncomment if using +# auto-import. +# .idea/artifacts +# .idea/compiler.xml +# .idea/jarRepositories.xml +# .idea/modules.xml +# .idea/*.iml +# .idea/modules +# *.iml +# *.ipr + +# CMake +cmake-build-*/ + +# Mongo Explorer plugin +.idea/**/mongoSettings.xml + +# File-based project format +*.iws + +# IntelliJ +out/ + +# mpeltonen/sbt-idea plugin +.idea_modules/ + +# JIRA plugin +atlassian-ide-plugin.xml + +# Cursive Clojure plugin +.idea/replstate.xml + +# SonarLint plugin +.idea/sonarlint/ +.idea/sonarlint.xml # see https://community.sonarsource.com/t/is-the-file-idea-idea-idea-sonarlint-xml-intended-to-be-under-source-control/121119 + +# Crashlytics plugin (for Android Studio and IntelliJ) +com_crashlytics_export_strings.xml +crashlytics.properties +crashlytics-build.properties +fabric.properties + +# Editor-based HTTP Client +.idea/httpRequests +http-client.private.env.json + +# Android studio 3.1+ serialized cache file +.idea/caches/build_file_checksums.ser + +# Apifox Helper cache +.idea/.cache/.Apifox_Helper +.idea/ApifoxUploaderProjectSetting.xml \ No newline at end of file diff --git a/.idea/.gitignore b/.idea/.gitignore new file mode 100644 index 0000000..26d3352 --- /dev/null +++ b/.idea/.gitignore @@ -0,0 +1,3 @@ +# Default ignored files +/shelf/ +/workspace.xml diff --git a/.idea/Homework4-Submission.iml b/.idea/Homework4-Submission.iml new file mode 100644 index 0000000..d8b3f6c --- /dev/null +++ b/.idea/Homework4-Submission.iml @@ -0,0 +1,8 @@ + + + + + + + + \ No newline at end of file diff --git a/.idea/inspectionProfiles/profiles_settings.xml b/.idea/inspectionProfiles/profiles_settings.xml new file mode 100644 index 0000000..105ce2d --- /dev/null +++ b/.idea/inspectionProfiles/profiles_settings.xml @@ -0,0 +1,6 @@ + + + + \ No newline at end of file diff --git a/.idea/misc.xml b/.idea/misc.xml new file mode 100644 index 0000000..8d86bfd --- /dev/null +++ b/.idea/misc.xml @@ -0,0 +1,7 @@ + + + + + + \ No newline at end of file diff --git a/.idea/modules.xml b/.idea/modules.xml new file mode 100644 index 0000000..e7fe12b --- /dev/null +++ b/.idea/modules.xml @@ -0,0 +1,8 @@ + + + + + + + + \ No newline at end of file diff --git a/.idea/vcs.xml b/.idea/vcs.xml new file mode 100644 index 0000000..35eb1dd --- /dev/null +++ b/.idea/vcs.xml @@ -0,0 +1,6 @@ + + + + + + \ No newline at end of file diff --git a/Class 4 Homework.ipynb b/Class 4 Homework.ipynb index 39bdb0b..f516b0b 100644 --- a/Class 4 Homework.ipynb +++ b/Class 4 Homework.ipynb @@ -80,19 +80,81 @@ "outputs": [], "source": [ "import fitz # PyMuPDF\n", + "from typing import List\n", + "import urllib, urllib.request\n", + "import feedparser\n", + "import requests\n", + "from io import BytesIO\n", + "import ssl, certifi, urllib.request\n", "\n", - "def extract_text_from_pdf(pdf_path: str) -> str:\n", + "context = ssl.create_default_context(cafile=certifi.where())\n", + "\n", + "def ssl_read_url(url: str) -> str:\n", + " return urllib.request.urlopen(url, context=context).read()\n", + "\n", + "def get_pdf_urls() -> List[str]:\n", + " url = f\"https://export.arxiv.org/api/query?search_query=all:electron&start=0&max_results=50\"\n", + " data = ssl_read_url(url)\n", + " res = feedparser.parse(data)\n", + " return [\n", + " link.href\n", + " for entry in res.entries\n", + " for link in entry.links\n", + " if \"pdf\" in link.href\n", + " ]\n", + "\n", + " \n", + "def extract_text_from_url(url: str) -> str:\n", " \"\"\"\n", " Open a PDF and extract all text as a single string.\n", " \"\"\"\n", - " doc = fitz.open(pdf_path)\n", + " response = requests.get(url)\n", + "\n", + " pdf_bytes = BytesIO(response.content)\n", + " doc = fitz.open(stream=pdf_bytes, filetype=\"pdf\")\n", " pages = []\n", " for page in doc:\n", " page_text = page.get_text() # get raw text from page\n", " # (Optional) clean page_text here (remove headers/footers)\n", " pages.append(page_text)\n", " full_text = \"\\n\".join(pages)\n", - " return full_text\n" + " return full_text" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50\n" + ] + } + ], + "source": [ + "urls = get_pdf_urls()\n", + "print(len(urls))" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50\n" + ] + } + ], + "source": [ + "full_texts = [extract_text_from_url(url) for url in urls]\n", + "print(len(full_texts))" ] }, { @@ -104,7 +166,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 36, "metadata": {}, "outputs": [], "source": [ @@ -115,7 +177,33 @@ " for i in range(0, len(tokens), step):\n", " chunk = tokens[i:i + max_tokens]\n", " chunks.append(\" \".join(chunk))\n", - " return chunks\n" + " return chunks\n", + "\n", + "chunks = [\n", + " chunk \n", + " for text in full_texts \n", + " for chunk in chunk_text(text)\n", + "]\n", + "\n", + "with open(\"chunks.json\", \"w\", encoding=\"utf-8\") as f:\n", + " json.dump(chunks, f, ensure_ascii=False, indent=4)" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "['Wiley-Interscience Publication, p61. [11] E. Benedetto et al., Proceedings of PAC07, p. 4033 (2007).', 'Proceedings of PAC07, p. 4033 (2007).', 'parallel plates separated by 0.2 mm is depicted. The electric field, driving the electron, 0 0.05 0.1', 'ctron and Positron Accelerators (MBI97), KEK, Tsukuba, Japan, 15-18 July 1997, KEK Proceedings 97-17', 'build up sim- ulations at CERN, in Proc.of ECLOUD12, La Biodola, Isola d’Elba, Italy, 5-9 June 2012.']\n" + ] + } + ], + "source": [ + "print([chunk[100:] for chunk in chunks[-5:]])" ] }, { @@ -128,19 +216,19 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 30, "metadata": {}, "outputs": [], "source": [ + "from sentence_transformers import SentenceTransformer\n", + "from numpy import ndarray\n", "\n", - "def chunk_text(text: str, max_tokens: int = 512, overlap: int = 50) -> List[str]:\n", - " tokens = text.split()\n", - " chunks = []\n", - " step = max_tokens - overlap\n", - " for i in range(0, len(tokens), step):\n", - " chunk = tokens[i:i + max_tokens]\n", - " chunks.append(\" \".join(chunk))\n", - " return chunks\n" + "model = SentenceTransformer(\"all-MiniLM-L6-v2\")\n", + "\n", + "def get_embeddings(chunks: List[str]) -> ndarray:\n", + " return model.encode(chunks) \n", + "\n", + "embeddings = get_embeddings(chunks)" ] }, { @@ -153,11 +241,11 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 37, "metadata": {}, "outputs": [], "source": [ - "\n", + "import json\n", "import faiss\n", "import numpy as np\n", "\n", @@ -167,10 +255,31 @@ "index.add(np.array(embeddings)) # add all chunk vectors\n", "\n", "# Example: search for a query embedding\n", - "query_embedding = ... # get embedding for the query (shape: [1, dim])\n", + "queries = [\"How do electron holes in materials work\",\n", + " \"Explain how spin-background affects low-lying excitations\",\n", + " \"Explain the properties of metallic adatoms\",\n", + " \"Explain hole-spin qubits\",\n", + " \"Explain why it's hard to scale up quantum computers\"\n", + " ]\n", + "query_embeddings = model.encode(queries) # get embedding for the query (shape: [1, dim])\n", "k = 3\n", - "distances, indices = index.search(query_embedding, k)\n", - "# indices[0] holds the top-k chunk indices\n" + "distances, indicesList = index.search(query_embeddings, k)\n", + "indicesList = indicesList.tolist()\n", + "retrieval_report = [\n", + " {\n", + " \"request\": queries[i],\n", + " \"response\": [chunks[index] for index in indices]\n", + " }\n", + " for i, indices in enumerate(indicesList)\n", + "]\n", + "\n", + "import pickle\n", + "\n", + "with open(\"index.pkl\", \"wb\") as f:\n", + " pickle.dump(index, f)\n", + "\n", + "with open(\"retrieval_report.json\", \"w\", encoding=\"utf-8\") as f:\n", + " json.dump(retrieval_report, f, ensure_ascii=False, indent=4)" ] }, { @@ -233,10 +342,24 @@ } ], "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, "language_info": { - "name": "python" + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.13.9" } }, "nbformat": 4, - "nbformat_minor": 2 + "nbformat_minor": 4 } diff --git a/README.md b/README.md new file mode 100644 index 0000000..697c1ad --- /dev/null +++ b/README.md @@ -0,0 +1,16 @@ +* Code Notebook / Script: See Class 4 Homework.ipynb +* Data & Index: See src/chunks.json & +* Retrieval Report: See retrieval_report.json +* FastAPI Service: The FastAPI app code (e.g. main.py) and instructions on how to run it. The /search endpoint should be demonstrable (e.g. returning top-3 passages in JSON for sample queries). + +# Running FastApi code +This model was trained on data about electrons. Queries should be related to electrons. + +Go to src and run uvicorn +```bash +cd src +uvicorn main:app +``` + + +Thank you to arXiv for use of its open access interoperability. \ No newline at end of file diff --git a/retrieval_report.json b/retrieval_report.json new file mode 100644 index 0000000..f00bbbe --- /dev/null +++ b/retrieval_report.json @@ -0,0 +1,42 @@ +[ + { + "request": "How do electron holes in materials work", + "response": [ + "positively charged defects and excess electrons · the positively charged defects create a surface-to-bulk electrostatic field and potential · the electrostatic field/potential induces band bending · the steep band bending depth profile can confine any charge carriers · the formerly unoccupied t2g conduction band follows the profile of the electrostatic potential and bends be- low EF in the near-surface region, thus giving rise to conduction electrons · these extra conduction electrons are due to the excess electrons from the oxygen vacancies · the conduction electrons see the depth profile of the electrostatic potential as a quantum well along the sur- face normal and they become confined along this direc- tion · as the surface normal is the only direction of confine- ment, conductions electrons are free to move only along a two-dimensional plane: in other words they form a 2DES · the spectroscopic fingerprint of the 2DES are subbands with appreciable energy-momentum dispersion along any direction parallel to the surface plane, but at the same time with well-determined energy values due to quantum confinement along the surface normal 5 The studies that signalled the discovery of a 2DES on STO(100) [21, 22] were however only the spark. Later research built on this knowledge and brought the field into a more profound understanding. We will summarize these results giving emphasis to ARPES studies that followed various research paths: (i) the experimental observation of 2DES characteristics that remained elusive (e.g. many-body interactions, spin polarization, further details on the creation mechanism, on energy dispersion and on dimensionality). (ii) the investigation of STO surfaces with different orienta- tions. (iii) the generation of 2DES on different functional oxides. (iv) the demonstration of a device-friendly fabrication pro- tocol that may bridge the gap between ARPES studies and technological applications. B. 2DES at the bare surface of SrTiO3: towards a more detailed understanding In order to shed more light into the 2DES, the role of irra- diation in creating a metallic surface was further investigated. Plumb and coworkers observed that on increasing irradiation, the size of the experimental Fermi surface (and hence the car- rier density) saturates very quickly, while intensity continues increasing. They proposed that increasing intensity signals a larger portion of the sample becoming metallic and conclude that there are two stable configurations of STO(100): one in- sulating (no irradiation, no 2D carriers, no Fermi surface) and one with a fixed density of metallic 2D carriers [38]. A later work by McKeown Walker et al. showed that there are not only two stable electronic configurations of STO(100) but that n2D can be continuously tuned as a function of the irradiation time (fluence) and the photon energy [39]. The authors sug- gested a mechanism of vacancy creation based on early results on core hole decay via an Auger process [40, 41]. The mecha- nism involves five consecutive steps: (i) an impinging photon creates a Ti3p core hole, (ii) the core hole is filled by O2p elec- trons via an Auger process, (iii) after the removal of electrons O2−can change to O+, (iv) Coulomb repulsion from neigh- boring Ti4+", + "following equations: (1) , (2) where the electric potential of electron cloud is given by . (3) Bunches, which are rigid Gaussian shape in transverse and are located equal spacing along s, are represented by their center of mass xp. FG is expressed by the Bassetti- Erskine formula. The electron cloud build-up is simulated by integrating the second equation for the motion of macro-electrons xe under xp=0. The initial condition of electrons, where and when electrons are created, is sketched in Figure 1. When beam pass through the chamber cross-section, electrons are created with an energy distribution. When an electron hits the wall, secondary electrons are produced with a probability. Figure 1: Electron cloud build-up model. The coupled bunch instability is studied by solving the two coupled equations (1) and (2). Coupled bunch instability in KEK-PF Progress of electron cloud effects in these 17 years (since 1995) started from the interpretation of an instability observed in KEK-Photon Factory. Very strong coupled-bunch instability had been observed since the positron storage had started. The threshold of the instability is very low 10-15mA. Either of positron or electron could be stored by changing the polarity of the magnets. The instability is observed only in positron storage. Figure 2 shows frequency spectrum for the instability published in [1]. The first electron cloud build- up code (PEI) is developed in 1995 [3], and the coupled bunch instability was interpreted as a wake effect of the electron cloud. The wake force was estimated by perturbation of the cloud due to a passage through of a shifted bunch [3]. Figure 3 shows the simulated wake force. The shape of wake force contains characteristic of the electron motion in the drift space. The wake force is the same direction as the shift of the bunch: namely, the bunch shift upper induces the wake force, by which following bunches near the shifted bunch are kicked upper. Namely, an upper shift of bunch, which induces electron motion toward upper direction, results upper force for following bunches. This feature is explained in Figure 4. This wake has the regular property as seen in [10]. Note the definition of the sign of the wake force is different. Growth rate for each mode is calculated by the wake force as follows, (4) The positive imaginary part of the frequency (Ωm) is the growth rate for m-th mode. Figure 5 shows unstable mode spectrum estimated by the simulated wake force. The growth rate is very fast, 0.3msec for m=250-300. The wake force (ω         −       − + = =    < ≤ < = E E E E E E E E E E E E b θ (2.9) (where ?(x) is the Heaviside theta function) with the only difference between them being the threshold energy E0, i.e. the energy value at which the metric parameters are constant, i.e. the metric becomes Minkowskian ( ?µ?(E= E0) = gµ? = diag(1,-1,-1,-1)); the fits to the experimental data", + "b θ (2.9) (where ?(x) is the Heaviside theta function) with the only difference between them being the threshold energy E0, i.e. the energy value at which the metric parameters are constant, i.e. the metric becomes Minkowskian ( ?µ?(E= E0) = gµ? = diag(1,-1,-1,-1)); the fits to the experimental data yield E0,e.m. = (4.5± 0.2) µeV; (2.10) E0,weak =(80.4± 0.2) GeV; ) ( ) ( ) ( 2 0 2 2 2 E b E b mc E mu E i i NR = = 6 Notice that for either interaction the metric is isochronous, spatially isotropic and ''sub- Minkowskian'', i.e. it approaches the Minkowskian limit from below (for EE0,weak > 80 GeV, and then our formalism is fully consistent with electroweak unification, which occurs at an energy scale ~ 100 GeV. Let us recall that the phenomenological electromagnetic metric (2.8)-(2.10) was derived by analyzing the propagation of evanescent waves in undersized waveguides(17). It allows one to account for the observed superluminal group speed in terms of a nonlocal behavior of the waveguide, just described by an effective deformation of space-time in its reduced part(5). As to the weak metric, it was obtained by fitting the data on the meanlife of the meson K0 S (experimentally known in a wide energy range (30÷350 GeV)(17)), thus accounting for its apparent departure from a purely Lorentzian behavior(3,21). 2) For the strong interaction, the metric was derived(4) by analyzing the phenomenon of Bose-Einstein (BE) correlation for p -mesons produced in high-energy hadronic collisions(18). Such an approach permits to describe the BE effect as the decay of a ''fireball'' whose lifetime and space sizes are directly related to the metric coefficients b2µ,strong(E), and to avoid the introduction of ''ad hoc'' parameters in the pion correlation function(4). The strong metric reads ( )) ( ), ( ), ( ), ( ) ( 2 ,3 2 ,2 2 ,1 2 ,0 E b E b E b E b diag E strong strong strong strong strong − − − = η (2.11) ( ) 0 , 1 ) ( 1 , / 0 ,1 ) ( ) ( ,0 5 2 ) ( 5 2 ) ( 2 ,0 ,0 , 0 2 , 0 , 0 2 , 3 2 , 0 2 2 , 2 2 2 , 1 >         −       − + = =    < ≤ < = = > ∀            =       = E E E E E E E E E E E E b E b E E b E b strong strong strong strong strong strong strong strong strong θ (2.12) with E0,strong = ( 367.5± 0.4) GeV . (2.13) Let us stress that, in this case, contrarily to the electromagnetic and the weak ones, a deformation of the time coordinate occurs; moreover, the three-space is anisotropic, with two spatial parameters", + "strong strong strong strong strong strong strong strong θ (2.12) with E0,strong = ( 367.5± 0.4) GeV . (2.13) Let us stress that, in this case, contrarily to the electromagnetic and the weak ones, a deformation of the time coordinate occurs; moreover, the three-space is anisotropic, with two spatial parameters constant (but different in value) and the third one va riable with energy like the time one. 7 3) The gravitational energy-dependent metric was obtained(6) by fitting the experimental data on the relative rates of clocks in the Earth gravitational field(20). Its explicit form is3: ( )) ( ), ( ), ( ), ( ) ( 2 ,3 2 ,2 2 ,1 2 ,0 E b E b E b E b diag E grav grav grav grav grav − − − = η (2.14) 0 , 1 1 4 1 ) ( 1 , 1 4 1 0 ,1 ) ( ) ( 2 , 0 , 0 ,0 2 , 0 , 0 2 , 3 2 , 0 >         −       + − + = =     <      + ≤ < = = E E E E E E E E E E E E b E b grav grav grav grav grav grav grav θ (2.15) with E0,grav = (20.2± 0.1) µeV . (2.16) Intriguingly enough, this is approximately of the same order of magnitude of the thermal energy corresponding to the 2.7°K cosmic background radiation in the Universe 4. Notice that the strong and the gravitational metrics are over-Minkowskian (namely, they approach the Minkowskian limit from above (E0E0,e.m.). On the basis of the previous considerations, it seems reasonable to assume that the physical object (particle) p with a rest energy (i.e. gravitational mass) just equal to the threshold energy E0,grav , namely E0,grav = mg,pc2 (4.3) must play a fundamental role for either e.m. and gravitational interaction. We can e.g. hypothesize that p corresponds to the lightest mass eigenstate which experiences both force fields (i.e., from a quantum viewpoint, coupling to the respective interaction carriers, the photon and the graviton). As a consequence, p must be intrinsically stable, due to the impossibility of its decay in lighter mass eigenstates, even in the case such a particle is subject to weak interaction, too (i.e. it couples to all gauge bosons of the Glashow-", + "respective interaction carriers, the photon and the graviton). As a consequence, p must be intrinsically stable, due to the impossibility of its decay in lighter mass eigenstates, even in the case such a particle is subject to weak interaction, too (i.e. it couples to all gauge bosons of the Glashow- Weinberg-Salam group SU(2) × U(1), not only to its electromagnetic charge sector). Since, as we have seen, for E = E0,grav the electromagnetic metric is minkowskian, too, it is natural to assume, for p: min,p,e.m. = min,p (4.4) namely its inertial mass is that measured with respect to the electromagnetic metric. Then, due to the Equivalence Principle (see Eq. (3.4)), the mass of p is characterized by    = = p in p g m m m m p , p,e.m. in, , grav p, in, : (4.5) 11 Therefore, for such a fundamental particle the LLI breaking factor (3.3) of the e.m. interaction becomes: ( ) . . , , . , . , . , , . , . . 1 1 m e p in p g p in p g p in p g p in m e m m m m m m m δ δ − = ⇔ − = − ≡ (4.6) Replacing (4.3) in (4.6) yields: ( ) . . 2 ,0 , 2 . . , ,0 1 1 1 m e grav p in m e p in grav c E m c m E δ δ − = ⇔ − = (4.7) Eq. (4.7) allows us to evaluate the inertial mass of p from the knowledge of the electromagnetic LLI breaking parameter de.m. and of the threshold energy E0,grav of the gravitational metric. On account of Eq. (3.1), we can relate the lowest limit to the LLI breaking factor of electromagnetic interaction, Eq. (3.3) (determined by the coil-charge experiment), with de.m as follo ws: d =1 - de.m. ≅ 4·10-11 (4.8) Then, inserting the value (2.16) for E0,grav 6 and (4.8) in (4.7), we get e in m e grav p in m c MeV c eV c E m , 2 2 11 5 . . 2 ,0 , 5.0 10 4 10 2 1 1 = = × × ≥ − = − − δ (4.9) (with min,e being the electron mass) where the ≥ is due to the fact that in general the LLI breaking factor constitutes an upper limit (i.e. it sets the scale under which a violation of LLI is expected). If experiment [25] does indeed provide evidence for a LLI breakdown (as it seems the case, although further confirmation is needed), Eq. (4.9) yields min,p=min,e. We find therefore the amazing result that the fundamental particle p is nothing but the electron e- (or its antiparticle e+ 7). The electron is indeed the lightest massive lepton (pointlike, non-composite particle) with electric charge, and therefore subjected to gravitational, electromagnetic and weak interactions, but unable to weakly decay due to its 6 Let us recall that the value of E0,grav was determined by fitting the experimental", + "its antiparticle e+ 7). The electron is indeed the lightest massive lepton (pointlike, non-composite particle) with electric charge, and therefore subjected to gravitational, electromagnetic and weak interactions, but unable to weakly decay due to its 6 Let us recall that the value of E0,grav was determined by fitting the experimental data on the slowing down of clocks in the Earth gravitational field (20) . See Ref. [6]. 7 Of course, this last statement does strictly holds only if the CPT theorem maintains its validity in the DSR framework, too. Although this problem has not yet been addressed in general on a formal basis, we can state that it holds true in the case we considered, since we assumed that the energy value is E = E0,grav corresponding to the Minkowskian form of both electromagnetic and gravitational metric. 12 small mass. Consequently, e- (e+) shares all the properties we required for the particle p, whereby it plays a fundamental role for gravitational and electromagnetic interactions. 5 – Mass relativity in DSR The considerations carried out in the previous Sections allow us therefore to relate the electron mass to the (local) breakdown of Lorentz invariance. Such a mass would then be a measure of the deviation of metric from the Minkowskian one. The minimum measured mass of a particle would be related to the minimum possible metric deviation compatible with its interactions. Such a point can be reinforced by the following argument. The maximal causal velocity ur defined by Eq. (2.3) can be interpreted, from a physical standpoint, as the speed of the quanta of the interaction locally (and phenomenologically) described in terms of a deformed Minkowski space. Since these quanta are associated to lightlike world-lines in 4 ~M , they must be zero-mass particles (with respect to the interaction considered), in analogy with photons (with respect to the e.m. interaction) in the usual SR. Let us clarify the latter statement. The carriers of a given interaction propagating with the speed ur typical of that interaction are actually expected to be strictly massless only inside the space whose metric is determined by the interaction considered. A priori, nothing forbids that such “deformed photons'' may acquire a non-vanishing mass in a deformed Minkowski space related to a different interaction. This might be the case of the massive bosons W+ , W- and Z0, carriers of the weak interaction. They would therefore be massless in the space 4 ~ M (ηweak(E)) related to the weak interaction, but would acquire a mass when considered in the standard Minkowski space M of SR (that, as already stressed, is strictly connected to the electromagnetic interaction ruling the operation of the measuring devices). In this framework, therefore, it is not necessary to postulate a ''symmetry breaking'' mechanism (like the Goldstone one in gauge theories) to let particles acquire mass. On the contrary, if one could build up measuring devices based on interactions different from the e.m. one, the photon might acquire a mass with respect to such a non-electromagnetic background. Mass itself would therefore assume a relative nature, related not", + "one in gauge theories) to let particles acquire mass. On the contrary, if one could build up measuring devices based on interactions different from the e.m. one, the photon might acquire a mass with respect to such a non-electromagnetic background. Mass itself would therefore assume a relative nature, related not only to the interaction concerned, but also to the metric background where one measures the energy of the physical system considered. This can be seen if one takes into account the fact that in general, for relativistic particles, mass is the invariant norm of 4-momentum, and what is usually measured is not the value of such an invariant, but of the related energy. 13 5 - Conclusions The formalism of DSR describes -among the others -, in geometrical terms (via the energy-dependent deformation of the Minkowski metric) the breakdown of Lorentz invariance at local level (parametrized by the LLI breaking factor dint). We have shown that within DSR it is possible - on the basis of simple and plausible assumptions - to evaluate the inertial mass of the electron e- (and therefore of its antiparticle, the positron e+) by exploiting the expression of the relativistic energy in the deformed Minkowski space + ∈0 ) ( ~ 4 R E E M , the explicit form of the phenomenological metric describing the gravitational interaction (in particular its threshold energy), and the LLI breaking parameter for the electromagnetic interaction de.m.. Therefore, the inertial properties of one of the fundamental constituents of matter and of Universe do find a ''geometrical'' interpretation in the context of DSR, by admitting for local violations of standard Lorentz invariance. We have also put forward the idea of a relativity of mass, namely the possible dependence of the mass of a particle on the metric background where mass measurements are carried out. 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Mignani: ''Boosts in an arbitrary direction and maximal causal velocities in a deformed Minkowski space'', Found. Phys. Lett. 16, 163 (2003) ), hep-th/0505032. 23. See C.M.Will: ''Theory and Experiment in Gravitational Physics'' (Cambridge Univ.Press, rev.ed.1993), and references therein. 24. U. Bartocci and M. Mamone Capria: Am. J. Phys. 59, 1030 (1991); Found. Phys. 21, 787 (1991). 25. F.Cardone and R.Mignani: Physics Essays 13, 643 (2000); ''On possible experimental evidence for a breakdown of local Lorentz invariance'' , in ''Gravitation, Electromagnetism and Cosmology: Toward a New Synthesis'' (Proc. Int. Conf. On Redshifts and Gravitation in a Relativistic Universe, Cesena, Italy, Sept. 17-20, 1999), ed. by K. Rudnicki (Apeiron, Montreal, 2001), p. 165; U.Bartocci, F.Cardone and R.Mignani: Found. Phys. Lett. 14, 51 (2001). 26. A. Marrani: ''Simmetrie di Killing di Spazi di Minkowski generalizzati'' (''Killing Symmetries of Generalized Minkowski Spaces'') (Laurea Thesis), Rome, October 2001 (in Italian). 27. F. Cardone, A. Marrani and R. Mignani: ''Killing symmetries of generalized Minkowski spaces. 2- Finite structure of space-time rotation groups in four dimensions'', Found. Phys. 34, 1155 (2004) , hep-th/0505105. 28. F. Cardone, A. Marrani and R. Mignani: ''Killing symmetries of generalized Minkowski spaces. 3- Space-time translations in four dimensions'', Found. Phys. 34, 1407 (2004), hep-th/0505116 .", + "Ultrafast Electron Dynamics in the Topological Insulator Bi2Se3 Studied by Time-Resolved Photoemission Spectroscopy J. A. Sobota,1, 2, 3 S.-L. Yang,1, 2, 3 D. Leuenberger,1, 2 A. F. Kemper,4 J. G. Analytis,5 I. R. Fisher,1, 2 P. S. Kirchmann,1, ∗T. P. Devereaux,1, 2 and Z.-X. Shen1, 2, 3, † 1Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA 2Geballe Laboratory for Advanced Materials, Department of Applied Physics, Stanford University, Stanford, CA 94305, USA 3Department of Physics, Stanford University, Stanford, CA 94305, USA 4Lawrence Berkeley National Lab, 1 Cyclotron Road, Berkeley, CA 94720 5Department of Physics, University of California, Berkeley, California 94720, USA (Dated: February 7, 2014) We characterize the topological insulator Bi2Se3 using time- and angle- resolved photoemission spectroscopy. By employing two-photon photoemission, a complete picture of the unoccupied elec- tronic structure from the Fermi level up to the vacuum level is obtained. We demonstrate that the unoccupied states host a second, Dirac surface state which can be resonantly excited by 1.5 eV photons. We then study the ultrafast relaxation processes following optical excitation. We find that they culminate in a persistent non-equilibrium population of the first Dirac surface state, which is maintained by a meta-stable population of the bulk conduction band. Finally, we perform a temperature-dependent study of the electron-phonon scattering processes in the conduction band, and find the unexpected result that their rates decrease with increasing sample temperature. We de- velop a model of phonon emission and absorption from a population of electrons, and show that this counter-intuitive trend is the natural consequence of fundamental electron-phonon scattering pro- cesses. This analysis serves as an important reminder that the decay rates extracted by time-resolved photoemission are not in general equal to single electron scattering rates, but include contributions from filling and emptying processes from a continuum of states. PACS numbers: 78.47.J-, 73.20.-r, 72.25.Fe, 79.60.-i,79.60.Bm INTRODUCTION Three dimensional topological insulators (TIs) are fas- cinating materials characterized by an insulating bulk electronic band structure with a metallic, conductive sur- face state (SS). The SS can be described by a Dirac equation for massless fermions, and is guaranteed to cross the band gap separating the bulk valence band (VB) and conduction band (CB) [1–5]. In addition, the SS is strongly spin-polarized with the electrons’ spin- orientations locked perpendicular to their momenta [6, 7]. This novel spin texture results in suppressed backscatter- ing [8] and makes the materials attractive for spintronics applications [9, 10]. Because of these intriguing properties, TIs have been the subject of intense investigation using ultrafast time- resolved techniques. These studies aim to unveil the fun- damental scattering properties of electrons in TIs by re- solving their dynamical processes in real time. Ultrafast optical techniques monitor transient changes in the op- tical properties of a material, such as reflectivity. These studies have yielded valuable insights on coherent phonon generation [11–13], spatial diffusion of photoexcited car- riers [12], and the timescales for electron-phonon scatter- ing [11–14]. A shortcoming of these techniques is that they lack momentum and energy resolution, and are lim- ited", + "properties of a material, such as reflectivity. These studies have yielded valuable insights on coherent phonon generation [11–13], spatial diffusion of photoexcited car- riers [12], and the timescales for electron-phonon scatter- ing [11–14]. A shortcoming of these techniques is that they lack momentum and energy resolution, and are lim- ited in their ability to directly distinguish surface and bulk effects. Time- and angle- resolved photoemission spectroscopy (trARPES) overcomes these limitations by adding momentum and energy resolution, allowing the material’s response to be studied directly within its elec- tronic band structure. With these capabilities, trARPES has provided deeper understanding on issues such as bulk-to-surface scattering [15–18], electron-phonon cou- pling [15–20], unoccupied electronic structure [21, 22], and novel topological states of matter [23]. In this article we demonstrate the wealth of informa- tion that can be gained about TIs from trARPES experi- ments. In particular, we employ both one-photon photoe- mission spectroscopy (1PPE) and two-photon photoemis- sion spectroscopy (2PPE) to study the electronic struc- ture and dynamics of the prototypical TI Bi2Se3. We first show that 2PPE resolves the unoccupied electronic structure with unprecedented clarity. This allows us to unambiguously identify a second SS well above the Fermi level EF . We find that optical excitation of n-type Bi2Se3 with 1.5 eV photons drives a direct optical transition pre- cisely into this state, thereby laying the foundation for direct ultrafast optical coupling to a topological SS [22]. We continue by reporting the electron dynamics fol- lowing this 1.5 eV excitation using time-resolved 2PPE. For this study we utilize p-type Bi2Se3, which has a com- pletely unoccupied SS in equilibrium, thereby granting us a clear view of the electron relaxation through the SS. We resolve the rapid decay of electrons to lower energy states via inter- and intra-band phonon-mediated scat- arXiv:1401.3078v2 [cond-mat.mtrl-sci] 6 Feb 2014 2 min max -0.2 0.0 0.2 EF p-type k|| (Å -1) surface state valence band conduction band 0.3 0.2 0.1 0.0 -0.1 E-ED (eV) -0.2 0.0 0.2 EF n-type FIG. 1. The band structure of the topological insulator Bi2Se3 as measured by ARPES. The band filling is determined by the bulk doping level. In an n-type sample (left), the conduction band, surface state, and valence band are occupied. In a p- type sample (right), only the valence band is occupied. tering processes. Within 2 ps a meta-stable population forms at the CB edge. This population acts as an elec- tron reservoir which fills the SS with a steady supply of carriers for ∼10 ps. This persistent occupation of a metallic state is a unique situation, since the absence of a bandgap in metallic bands typically means that there is no barrier to rapid recombination. This long-lived spin- textured metallic population may present a channel in which to drive transient spin-polarized currents [15]. Finally, we examine the role of electron-phonon scat- tering processes by studying the dynamics of the CB as a function of sample temperature. We discover de- creasing population relaxation rates with increasing sam- ple temperature. As higher temperatures correspond to higher phonon occupations, this result might seem counter-intuitive", + "currents [15]. Finally, we examine the role of electron-phonon scat- tering processes by studying the dynamics of the CB as a function of sample temperature. We discover de- creasing population relaxation rates with increasing sam- ple temperature. As higher temperatures correspond to higher phonon occupations, this result might seem counter-intuitive at first glance. However, we show that this behavior is reproduced by a straightforward electron- phonon scattering model. While we aim to achieve only qualitative agreement in this work, this model has the potential to extract detailed, quantitative information about electron-phonon coupling from trARPES data. METHODS Single crystals of Bi2Se3 were synthesized as described in Ref. 15. p-type crystals were achieved by Mg-doping. Fig. 1 shows typical ARPES spectra for the samples used in this work. The difference in the doping levels of n− and p−type samples is clearly exhibited by the energetic position of EF and thus the band filling. We illustrate our setup for trARPES in Fig. 2(a). A Ti-Sapphire oscillator operating at 80 MHz, or regener- ative amplifier at 310 kHz, generates 820 nm (1.5 eV) infrared laser pulses which are split into two paths. In Frequency,x4 Laser Ti:Sapphire,oscillator,/,amplifier 80MHzg,50fs,/,310kHzg,35fs 820nm 205nm Optical,delay Δt Sample (a) (b) Energy Δt 2PPE 1PPE Evac EF Photoelectron,analyzer (i) (ii) (iii) FIG. 2. (a) The experimental setup used for the pump-probe photoemission experiments described in this work. A laser generates ultrafast laser pulses at 820 nm (1.5 eV). A portion of the beam is used to optically excite the sample, and the remaining beam is frequency quadrupled to 205 nm (6.0 eV) to probe by photoemission. A variable path length allows for a tunable delay between the two optical paths. (b) Schematic of the photoemission processes utilized in this work. (i) Monochromatic 2PPE using 6 eV photons, (ii) Bichromatic 2PPE using time-delayed 1.5 eV and 6 eV photons, and (iii) 1PPE using 6 eV photons. The photoelectrons are collected by a hemispherical photoelectron analyzer. one path the pulse frequency is quadrupled to produce a 205 nm (6.0 eV) ultraviolet beam. The other path in- cludes a delay stage which varies the optical path length. Both beams are focused on the sample in an ultrahigh vacuum chamber. By tuning the delay stage, the tempo- ral delay between both pulses can be varied. This setup permits three primary modes of measure- ment, as summarized in Fig. 2(b). Process (i) represents 2PPE performed with the 6 eV pulses. The entire two- photon process occurs within the pulse duration of the incident 6 eV pulse. Since no time delay is introduced, this is a static measurement. However, due to the finite time duration of the laser pulses (of order 100 fs in our case), electron relaxation processes can occur in the in- termediate states before the electron is photoemitted. As will be shown below, this mode of measurement grants access to the unoccupied electronic structure between EF and the vacuum level Evac [24–26]. Process (ii) represents time-resolved 2PPE performed with time-delayed 1.5 and 6 eV pulses. Here the 1.5 eV photon plays the role of a pump", + "is photoemitted. As will be shown below, this mode of measurement grants access to the unoccupied electronic structure between EF and the vacuum level Evac [24–26]. Process (ii) represents time-resolved 2PPE performed with time-delayed 1.5 and 6 eV pulses. Here the 1.5 eV photon plays the role of a pump by optically exciting electrons within the material’s band structure, while the 6 eV photon serves to probe the transiently modified elec- tron distribution by photoemission. A complete movie of 3 the excitation and relaxation dynamics is obtained by measuring the photoemission spectrum as a function of pump-probe delay [24–26]. Finally, process (iii) represents 1PPE performed with 6 eV pulses. This is equivalent to conventional ARPES, and permits measurement of the equilibrium, occupied electronic structure [27]. We note the simplicity and flexibility of the setup which is achieved by using the same 6 eV photon en- ergy for both 1PPE and 2PPE processes. Our exper- imental configuration is identical for all measurements, except for the incident photon intensities. Since 1PPE scales linearly with peak light intensity and 2PPE scales quadratically, we switch between 1PPE and 2PPE acqui- sition modes merely by tuning the intensity. This is a sig- nificant distinction between our measurement and stan- dard 2PPE measurements: Conventional 2PPE is per- formed with the photon energies chosen to be less than the sample work function, so that an overwhelmingly in- tense 1PPE signal is avoided. Here we deliberately use a photon energy larger than the work function, allowing us to perform both 1PPE and 2PPE with the same 6 eV pulses. To avoid the intense 1PPE signal while measur- ing in mode (i), we operate the analyzer such that only electrons with energy > EF are collected All measurements are performed in an ultrahigh vac- uum chamber with pressure maintained below 1×10−10 Torr to minimize sample aging. The emission angle and kinetic energy of the photoelectrons are resolved by a hemispherical electron analyzer. The setup has a total energy resolution of < 22 meV (limited by the bandwidth of the 6eV pulses) and angular resolution < 0.5◦. The temporal resolution, defined by the cross-correlation of the 1.5 and 6 eV pulses, is about 160 fs. The density functional theory calculations were done with the PBE exchange functional [28] using the full potential (linearized) augmented plane-wave method as implemented in the wien2k package[29]. The calcula- tions were based on the experimentally determined crys- tal structure reported in Ref. 30. The Bi2Se3 slab con- sists of 6 quintuple layers, separated by a vacuum of ∼25 ˚A. We used a k-mesh size of 15×15×1, and utilized spin-orbit coupling in the self-consistent calculations un- less otherwise indicated. RESULTS & DISCUSSION Optical coupling to a second Dirac surface state We begin by presenting the 2PPE and 1PPE spectra of p-type Bi2Se3 in the upper and lower panels of Fig. 3(a). These spectra correspond to processes (i) and (iii) of Fig. 2(b). The 1PPE spectrum consists of only a narrow band of intensity from the occupied portion of the VB above the low-energy cutoff, as in Fig. 1", + "spectra of p-type Bi2Se3 in the upper and lower panels of Fig. 3(a). These spectra correspond to processes (i) and (iii) of Fig. 2(b). The 1PPE spectrum consists of only a narrow band of intensity from the occupied portion of the VB above the low-energy cutoff, as in Fig. 1 [15]. The 2PPE spec- -0.2 0.2 -0.2 0.0 0.2 k|| dÅ -1) 1PPE 10 -3 10 -1 da) db) dc) Intensityodau) Γ M K 0 1 SurfaceoBiocontribution 6 5 4 3 2 1 0 Eo-oEF deV) Evac 2PPE { { 1 st SS 2 nd SS FIG. 3. (a) 2PPE and 1PPE spectra of p-type Bi2Se3. The 1PPE spectrum contains only a narrow band of intensity from the occupied portion of the VB above the low-energy cutoff. The 2PPE spectrum reveals states from EF up to Evac, in- cluding the well-known Dirac state around ∼0.5 eV, as well as a second Dirac state around ∼1.8 eV. The intensities are rescaled exponentially as a function of energy to make all fea- tures visible on the same image. (b) Momentum-integrated energy distribution curves showing the dynamic range of the 2PPE intensity. (c) Slab band structure calculation, which shows a one-to-one correspondence with the observed fea- tures. Figure reproduced from Ref. 22 (Copyright 2013 by the American Physical Society). trum, on the other hand, reveals states all the way from EF up to Evac. Just above EF we find the familiar CB and SS, which would be occupied in an n-type sample. Note that the spectral intensity spans a dynamic range of ∼3 orders of magnitude (see Fig. 3(b)) and so the intensities of 3(a) are rescaled exponentially with energy to allow all features to be displayed simultaneously. Before discussing these results in greater detail, it is important to understand the origin of the observed spec- tral features. In principle, the 2PPE spectral intensity can be complicated by contributions from both initial and final states, in addition to the intermediate states which are the subject of our study [25, 26]. To check whether this is the case, we show the calculated band structure in Fig. 3(c). While the energy scales are not reproduced exactly, there is an unambiguous one-to-one correspon- dence between the observed and calculated features. This demonstrates that the 2PPE spectrum is dominated by the intermediate states. It is not immediately clear why a strong dependence of the initial states is not observed. This is likely due to the fact that the electronic states populated by the 6 eV photons are very high in energy, which means there 4 is a large phase space for decay, with correspondingly short lifetimes (∼1 fs) [31]. We therefore expect signifi- cant inter-and intra- band scattering to occur within the duration of the 6 eV pulse. In effect, these relaxation processes rapidly redistribute electrons among all the un- occupied bands. Now that we understand the origin of the 2PPE spec- tral features, we proceed by discussing the electronic states located between 1.3 and 2.4 eV of Fig. 3(a). The band structure here is strikingly similar to the", + "effect, these relaxation processes rapidly redistribute electrons among all the un- occupied bands. Now that we understand the origin of the 2PPE spec- tral features, we proceed by discussing the electronic states located between 1.3 and 2.4 eV of Fig. 3(a). The band structure here is strikingly similar to the VB, CB, and SS bands near EF . In fact, Niesner et. al. first identified the linearly dispersive band as an unoccupied topologically protected, spin chiral SS, while the upper and lower bands are the corresponding bulk bands[21]. To distinguish these two sets of bands, we refer to the Dirac cone near EF as the 1st SS, and the higher en- ergy Dirac cone as the 2nd SS. The corresponding bulk bands are referred to as the 1st and 2nd VB and CB. The 2nd SS disperses with a velocity of 3.3 × 105 m/s at the Dirac point, which is comparable to the velocity of 5.4 × 105 m/s typically measured for the Fermi velocity of the 1st SS [32]. Calculations confirm the assignment of these bands to bulk and surface states [21, 22, 33]. Moreover, just like the 1st SS, the 2nd SS exists only in the presence of crystal spin-orbit interaction [21, 22, 33]. This suggests that both SSs share the same physical ori- gin, as they both arise due to symmetry inversion of bulk states in the presence of strong spin-orbit coupling. As we shall show, the existence of the 2nd SS could be highly relevant to studies utilizing ultrafast optical exci- tation of TIs. These experiments typically use a photon energy of 1.5 eV due to the use of a Ti:Sapphire laser source. Furthermore, n-type samples are more commonly used than p-type. We have therefore investigated the ef- fect of optically exciting n-type Bi2Se3 with 1.5 eV pho- tons. In Fig. 4(a) we first show for reference the unoccupied 2nd SS obtained by 6 eV 2PPE. The effect of 1.5 eV exci- tation is shown in Fig. 4(b). Specifically, we use 1.5 eV to optically excite the sample, and 6 eV to photoemit. This corresponds to process (ii) of Fig. 2(b), with the time delay set to zero. The corresponding 1PPE spectrum of the occupied band structure is shown in Fig. 4(c). A sim- ilar set of measurements, but for a sample with a slightly higher doping level, is shown in Figs. 4(d) and (e). The most striking feature in the spectra of Figs. 4(b) and (d) is a linearly dispersing feature. At first glance, this would appear to represent the 2nd SS, with features above and below representing the 2nd CB and VB. How- ever, the appearance of these spectra is highly mislead- ing: Comparison with Fig. 4(a) shows that this cannot be the 2nd SS, since the energies of the Dirac points do not match. We instead identify the linearly dispersing feature as 2PPE from the 1st SS. This assignment is sup- ported by the fact that its Dirac point is located precisely 1.5 eV above the Dirac point of the 1st SS. 0.1 -0.1", + "since the energies of the Dirac points do not match. We instead identify the linearly dispersing feature as 2PPE from the 1st SS. This assignment is sup- ported by the fact that its Dirac point is located precisely 1.5 eV above the Dirac point of the 1st SS. 0.1 -0.1 6eV 0.2 0.0 0.1 -0.1 6eV 1.5+6eV 1.5+6eV 2.0 1.8 1.6 1.4 1.2 0.1 -0.1 6+6eV E - ED (eV) k|| (Å -1) 1.5eV (a) (b) (c) (d) (e) (f) FIG. 4. Optical excitation of the 2nd Dirac cone using 1.5 eV photons on n-type Bi2Se3. Dashed lines are guides to the eye.(a) For reference, a 6 eV 2PPE spectrum of the 2nd Dirac cone is shown. (b) 2PPE spectrum using 1.5 eV photons to populate the 2nd SS, and 6 eV photons to photoemit. (c) 6 eV 1PPE measurement of the occupied states. Panels (d) and (e) are the same measurements, but for a sample with a higher level of n-type doping. (f) Cartoon picture of the 1.5 eV excitation, 6 eV probe process. The resulting 2PPE spectrum can be understood as a projection of the initial states onto the intermediate states. Both resonant and non-resonant 2PPE processes are resolved. Green shading represents the resonant processes. In general, the spectra of Figs. 4(b) and (d) can be un- derstood as a projection of the initial states (represented by (c) and (e)) onto the intermediate states 1.5 eV above (represented by (a)). The features are brightest where the initial and intermediate states overlap in energy and momentum. This is most dramatic in Fig. 4(b), where the projection of the upper half of the 1st SS onto the lower half of the 2nd SS results in two brights spots of intensity. These are known as resonant 2PPE processes, since the 1.5 eV photon resonantly excites a state 1.5 eV above the initial state at the same wave vector. The less intense features represent non-resonant 2PPE processes [25, 26]. A cartoon summary is provided in Fig. 4(f), where resonant 2PPE processes are highlighted in green. We note a significant difference between the 2PPE re- sults of Figs. 3 and 4. For Fig. 3 we emphasized that the dispersion of the initial states plays a negligible role in the appearance of the 2PPE spectrum, while the re- sults of Fig. 4 are a complicated combination of initial- and intermediate- state effects. While the reason for this difference is not fully understood, we believe it is re- lated to the fact that 1.5 eV photons promote electrons to much lower energy than 6 eV photons. This means there is substantially less phase space for decay of the photo-excited electrons, and correspondingly longer elec- tron lifetimes. Therefore, the rapid scattering processes which are responsible for washing out the initial state ef- 5 fects for 6 + 6 eV 2PPE are much less significant in the case of 1.5 + 6 eV 2PPE. Persistent surface state population We now use time-resolved 2PPE to investigate the ul- trafast dynamics of optically excited electrons in Bi2Se3. We confine our attention", + "the initial state ef- 5 fects for 6 + 6 eV 2PPE are much less significant in the case of 1.5 + 6 eV 2PPE. Persistent surface state population We now use time-resolved 2PPE to investigate the ul- trafast dynamics of optically excited electrons in Bi2Se3. We confine our attention to the 1st SS, CB, and VB. Full details of this experiment are reported in Ref. 15. For this study we utilize p-type samples, since the initially unoccupied SS provides for an exceptionally clear view of electron relaxation through the SS. An overview of the dynamics upon 1.5 eV excitation with 26 µJ/cm2 absorbed fluence is shown in Fig. 5(a): Starting in equilibrium (t < 0), the SS and CB are ini- tially unoccupied. Upon excitation, high-lying electronic states like those in Fig. 4 are populated. After ∼0.7 ps those electrons scatter to lower energy, populating the SS and CB. After ∼2.5 ps the SS and CB populations have significantly decayed and energetically relaxed to- wards the bottom of their respective bands. The subse- quent dynamics is much slower and persists for > 9 ps. Obstructed from further decay due to the bandgap, the CB electrons form a meta-stable population at the CB edge. Intriguingly, this CB population is accompanied by a persistent population in the SS, but only energetically below the CB edge. The observation of a long-lived CB population is not unusual in semiconductors [34]. The longevity of this population is due to the fact that the electron-phonon scattering processes responsible for energy relaxation within the band are incapable of recombining electrons with the VB. This is because the highest phonon energy in Bi2Se3 is ≈23 meV [35], while the bandgap is 200 meV [36]. The simultaneous persistence of the metallic SS population, however, is surprising since the absence of a bandgap means there is no barrier to rapid recombina- tion. We attribute this SS population to a continuous filling from the meta-stable CB population. Evidence for this can be found in the population dynamics. In Fig. 5(b) we plot the transient photoemission intensity obtained by integrating within the windows indicated in Fig. 5(c). Above the CB, the SS decays with a single exponential. However, below the CB a second slower component is observed, reflecting the presence of an ad- ditional filling channel. Moreover, the decay rate of this slower component matches that of the population decay at the CB edge (see curves #6 and #10). The agreement of these decay rates is direct evidence that the persistent SS population is due to continuous filling from the meta- stable CB population. A cartoon summarizing these pro- cesses is shown in Fig. 5(d). We note that similar bulk- to-surface scattering behavior has been observed in Si, which however lacks a metallic SS crossing EF [37, 38]. More recently, trARPES experiments on n-type Bi2Se3 -7 0 7 0.7ps -7 0 7 2.5ps 0.6 0.4 0.2 0.0 E-EF ueVD -7 0 7 -0.5ps -7 0 7 ux4D 9.0ps AngleoudegD AngleoudegD 0.6 0.4 0.2 0.0 E-EF ueVD to=o0 ~0.5ps ~2.5ps >5ps (a) 0.01 0.1", + "SS crossing EF [37, 38]. More recently, trARPES experiments on n-type Bi2Se3 -7 0 7 0.7ps -7 0 7 2.5ps 0.6 0.4 0.2 0.0 E-EF ueVD -7 0 7 -0.5ps -7 0 7 ux4D 9.0ps AngleoudegD AngleoudegD 0.6 0.4 0.2 0.0 E-EF ueVD to=o0 ~0.5ps ~2.5ps >5ps (a) 0.01 0.1 1 8 6 4 2 0 NormalizedoIntensityouauD DelayoupsD 1 2 3 4 5 6 7 8 9 10 1 9 10 (b) (c) (d) -7 0 7 FIG. 5. (a) Overview of the dynamics generated by 1.5 eV optical excitation of p-type Bi2Se3. The SS and CB are unoc- cupied in equilibrium, but are transiently populated and un- dergo subsequent decay processes. The intensity in the fourth panel is multiplied by a factor of 4 to allow weak features to be seen. (b) Transient photoemission intensities within the inte- gration windows indicated in the subsequent panel (c). Curve #10 corresponds to the integration window over the CB edge, and is normalized to match the intensity of the SS intensities. The slow component of the SS decay has the same 5.95(2) ps timescale as the CB population decay. (d) Schematic of the transitions and scattering processes generated by 1.5 eV exci- tation of p-type Bi2Se3, including the direct optical transition, scattering into SS and CB, intra-band scattering of the CB and SS, and the CB-to-SS scattering responsible for a persis- tent SS population. Figure adapted from Ref. 15. have also identified signatures of a bulk-to-surface scat- tering channel mediated by phonons [16–18]. An exponential fit to the CB population decay (curve #10 in Fig. 5(b)) gives a population lifetime of 5.95(2) ps. The decay mechanisms responsible for this lifetime may have several contributions. We have argued that 6 0.01 0.1 1 Norm.CIntensity 8 6 4 2 0 DelayC(ps) 4 3 2 1 0 DecayCRateC(ps -1) 0.15 0.10 0.05 0.00 E-ECB (eV) TempC(K) 40 200C (a) (b) (c) 0.50 0.40 0.30 0.20 RateCatCECB (ps -1) 200 150 100 50 TemperatureC(K) FIG. 6. (a) Transient photoemission intensity at a fixed en- ergy E −ECB =0.08 eV for temperatures from 40 to 200 K in 20 K steps. The population relaxation becomes slower for increasing temperature. (b) The extracted population decay rates as a function of temperature and energy. The rates at the CB edge (E = ECB) are marked with triangles and plot- ted as a function of temperature in (c). The smooth curve is a quadratic fit to the data points. electron-phonon coupling cannot be responsible for re- combination due to the low phonon energies. Direct re- combination via photon emission is a possibility, though the timescale for such a process is typically ≫1 ns, and thus seems unlikely to be relevant on our measurement timescale [39]. The CB decay could also be attributed to spatial diffusion of CB electrons away from the probed volume of the sample. A characteristic timescale for this process was estimated to be ∼21 ps[15]. Of course, the inter-band scattering channel into the SS provides an- other effective recombination mechanism. Future studies utilizing Bi2Se3 thin films to mitigate spatial", + "to spatial diffusion of CB electrons away from the probed volume of the sample. A characteristic timescale for this process was estimated to be ∼21 ps[15]. Of course, the inter-band scattering channel into the SS provides an- other effective recombination mechanism. Future studies utilizing Bi2Se3 thin films to mitigate spatial diffusion may be helpful in disentangling these processes. Energy relaxation via electron-phonon scattering in the conduction band We have argued that the optically excited electron pop- ulation in the CB relaxes via electron-phonon scattering processes. Since the phonon population changes with sample temperature, we expect the relaxation rates to exhibit a temperature dependence. Accordingly, we have performed temperature-dependent studies of the electron relaxation in the CB of p-type Bi2Se3. Fig. 6(a) shows the transient photoemission intensity at a fixed energy relative to the band edge ECB for tem- peratures from 40 to 200 K in 20 K steps using an ab- sorbed pump fluence of 1.3 µJ/cm2. We utilize this low fluence to minimize sample heating and thus have a well defined sample temperature. While the initial rise of the population does not exhibit any pronounced tempera- ture dependence, the subsequent decay slows down with increasing temperature. We fit a single exponential to the initial decay and plot the resulting decay rates as function of energy in Fig. 6(b). There are two distinct temperature dependencies: near ECB the rates increase with increasing temperature, and at higher energies the rates decrease with increasing temperature. To under- stand the origin of these two behaviors, we must distin- guish between two types of scattering processes: intra- band scattering, which conserves the electron number in the band, and band-emptying processes. The decay rate at E = ECB cannot represent intra-band scattering, since the phase space for decay at the band edge is strictly zero. Instead, the decay rate at ECB must represent a band- emptying rate. The emptying rate extracted from the data is plotted as a function of temperature in Fig. 6(c). We proceed to discuss the intra-band scattering pro- cesses that dominate away from ECB. The dramatic decrease of these rates with increasing sample tempera- ture seems counter-intuitive when one considers that the phonon population n increases with increasing tempera- ture T for all phonon energies ω, as represented by the Bose-Einstein distribution: [40] n(ω) = 1 eω/kBT −1 (1) This poses the question of what mechanism can result in decreasing rates as the temperature is increased. To address this question we model the change in electron distribution due to absorption and emission of phonons. The corresponding scattering rates follow from Fermi’s golden rule [40]. For simplicity, we neglect any momentum dependence in the following discussion. Un- der these assumptions, the rate for an electron to scatter out of the state at ϵ due to coupling with a phonon of energy ω is given by: Γabs(ϵ, ω) = 2π ℏ|G(ω)|2n(ω)N(ϵ + ω)(1 −f(ϵ + ω)) (2) Γemis(ϵ, ω) = 2π ℏ|G(ω)|2(1 + n(ω))N(ϵ −ω)(1 −f(ϵ −ω)) (3) Here Γabs (Γemis) denotes the rate attributable to phonon absorption (emission), N(ϵ) is the electronic den- sity", + "to coupling with a phonon of energy ω is given by: Γabs(ϵ, ω) = 2π ℏ|G(ω)|2n(ω)N(ϵ + ω)(1 −f(ϵ + ω)) (2) Γemis(ϵ, ω) = 2π ℏ|G(ω)|2(1 + n(ω))N(ϵ −ω)(1 −f(ϵ −ω)) (3) Here Γabs (Γemis) denotes the rate attributable to phonon absorption (emission), N(ϵ) is the electronic den- sity of states (DOS) and f(ϵ) is the electron occupation, which is a number between 0 and 1. G(ω) is the matrix element describing the scattering process. Physically, it encodes the electron coupling strength to phonons of en- ergy ω. In addition to scattering out of the state at ϵ, we must also consider scattering in from other states. There are therefore four processes which must be included, as repre- sented in Fig. 7. We introduce the phonon DOS F(ω) to 7 Energy (i) (ii) (iii) (iv) ε Γabs(ε-ω) Γemis(ε+ω) Γemis(ε) Γabs(ε) FIG. 7. Cartoon of the four electron-phonon scattering pro- cesses relevant for the relaxation of electrons in the CB. (i) and (ii) represent the filling of the state at energy ϵ by phonon emission and absorption, respectively. (iii) and (iv) represent the emptying of the state at energy ϵ by phonon absorption and emission, respectively. integrate over all ω, and obtain rate equations describing the time evolution of f(ϵ): N(ϵ)∂fout(ϵ) ∂t = Z dω F(ω)[Γabs(ϵ, ω)N(ϵ)f(ϵ) + Γemis(ϵ, ω)N(ϵ)f(ϵ)] (4) N(ϵ)∂fin(ϵ) ∂t = Z dω F(ω)[Γabs(ϵ −ω, ω)N(ϵ −ω)f(ϵ −ω) + Γemis(ϵ + ω, ω)N(ϵ + ω)f(ϵ + ω)] (5) The complete dynamics of the model are then de- scribed by: ∂f(ϵ) ∂t = ∂fin(ϵ) ∂t −∂fout(ϵ) ∂t −Γescf(ϵ) (6) The last term Γesc represents the rate at which elec- trons escape the band. This term can be set to zero to limit the calculation to intra-band processes only and conserve the electron number in the band. For the numerical calculation we shall assume (1 −f(ϵ)) ≈1 in Eqs. 2 and 3. This is not required for the calculation, but is a reasonable assumption for the CB of p-type Bi2Se3 since it is completely unoccu- pied in equilibrium, and only perturbatively populated by the optical excitation. The lattice temperature en- ters through the temperature dependence of the Bose- Einstein distribution n(ω). In this formulation we as- sume that the lattice temperature remains constant dur- ing the electron relaxation process. This assumption is consistent with experiments in which the lattice under- goes a much smaller transient change in temperature as compared to the electrons due to its significantly higher specific heat [41, 42]. Four quantities need to be specified to proceed with the calculation: N(ϵ), |G(ω)|2, F(ω), and f(ϵ)|t=0, which de- fines the initial condition for the evolution of the electron distribution. 0.001 0.01 0.1 1 f(ε) (arb. u.) 0.12 0.08 0.04 0.00 ε (eV) Time 0.0 3.6 100 2 4 1000 2 Temperature (K) 8 6 4 2 0 Time (arb. u.) 1.0 0.8 0.6 0.4 0.2 0.0 N(ε) (arb. u.) 0.12 0.08 0.04 0.00 ε (eV) 1.0 0.8 0.6 0.4 0.2 0.0 F(ω) (arb. u.) 18 12 6 0 ω (meV) (a) (b) (c) (d) FIG. 8. Results of the", + "2 Temperature (K) 8 6 4 2 0 Time (arb. u.) 1.0 0.8 0.6 0.4 0.2 0.0 N(ε) (arb. u.) 0.12 0.08 0.04 0.00 ε (eV) 1.0 0.8 0.6 0.4 0.2 0.0 F(ω) (arb. u.) 18 12 6 0 ω (meV) (a) (b) (c) (d) FIG. 8. Results of the electron-phonon scattering model com- puted with a phonon temperature of 100 K and Γesc = 0. (a) The electronic density of states and (b) the phonon density of states used as inputs for the calculation. (c) The com- puted electronic distribution as a function of time. (d) The extracted transient electronic temperature. For N(ϵ) we assume a linearly increasing DOS, as shown in Fig. 8(a). We base this assumption on a com- parison to the DOS measured by scanning tunneling ex- periments [43]. Note that we only specify N(ϵ) up to a proportionality factor. This implies that the tempo- ral units of our calculation will be arbitrary. Neverthe- less, this will be sufficient to understand the qualitative behavior. The choice of |G(ω)|2 is less straightforward. We found that the choice of |G(ω)|2 does not qualita- tively change the results. To avoid detailed calculations of electron-phonon coupling matrix elements, we shall assume that all phonon modes couple equally well to all electrons, so that |G(ω)|2 is simply a constant. The F(ω) used for our modeling is shown in Fig. 8(b). This choice is an approximation to the measured phonon DOS [44]. Finally, we can use our trARPES data to make a suitable choice for f(ϵ)|t=0. We use f(ϵ)|t=0 ∝e−ϵ/ϵ0 with ϵ0 = 0.2 eV. With these four assumptions, it is straightforward to compute the temporal evolution of f(ϵ) for any T and Γesc. The results for T = 100 K with Γesc = 0 are shown in Fig. 8(c). Interestingly, the electronic distribution remains essentially exponential as it evolves, and eventu- ally saturates to a steady-state solution. In Fig. 8(d) we extract a transient electronic temperature by fitting f(ϵ) to an exponential distribution at each time. Reassur- ingly, the electronic temperature settles at 100 K. This demonstrates that this simple model captures the pro- cess by which electrons reach thermal equilibrium with the lattice. In fact, it can be shown analytically that ∂fin ∂t = ∂fout ∂t for f ∝e−ϵ/T , where T is the lattice tem- perature. In other words, the steady-state electronic dis- tribution is a Boltzmann function at thermal equilibrium 8 (b) 4 3 2 1 0 0.15 0.10 0.05 0.00 ε (eV) Temp (K) 40 200 4 3 2 1 0 Decay Rate (arb. u.) 0.15 0.10 0.05 0.00 ε (eV) (a) FIG. 9. Temperature dependence of the electron-phonon scat- tering model. (a) The population decay rates for varying sam- ple temperature. Γesc is set to the temperature-dependent values extracted from the fit in Fig. 6(c). (b) The population decay rates, but with Γesc set to the temperature-independent value of 0.17. with the lattice. (Note that Fermi-Dirac statistics need not be considered here due to the assumption f ≪1 made in this analysis). Now that we understand the behavior", + "extracted from the fit in Fig. 6(c). (b) The population decay rates, but with Γesc set to the temperature-independent value of 0.17. with the lattice. (Note that Fermi-Dirac statistics need not be considered here due to the assumption f ≪1 made in this analysis). Now that we understand the behavior of this model for fixed T, we continue by varying the temperature. Be- cause the behavior of the rate curves of Fig. 6(b) are strongly affected by the temperature-dependence of the emptying rate at ECB, we set Γesc equal to the fitted val- ues in Fig. 6(c). In Fig. 9(a) we show the extracted exponential decay rate as a function of ϵ and T. The result agrees remarkably well with the experimental data of Fig. 6(b). In particular, away from the band edge it exhibits a monotonic decrease of relaxation rate with increasing T. In addition, the model reproduces the flat- tening of the rate curve at the value of Γesc near the band edge. While the T-dependent Γesc is required to reproduce the behavior of the experimental data near ECB, we find it instructive to repeat the calculation for a fixed, T-independent value of Γesc. The resulting rates are shown in Fig. 9(b). With the T-dependence of Γesc removed, only the temperature dependence of the intra- band electron-phonon scattering processes remains. This clearly demonstrates that the counter-intuitive trend of increasing decay rates with decreasing T can be at- tributed to fundamental electron-phonon scattering pro- cesses alone. CONCLUSIONS & OUTLOOK In summary, we have characterized the electronic structure of Bi2Se3 with a combination of 1PPE and 2PPE techniques, performed time-resolved measure- ments on p-type Bi2Se3, and elucidated the rich electron dynamics generated by optical excitation. In particular, we demonstrated that 2PPE with 6 eV photons illumi- nates the states between EF and Evac. This technique reveals a 2nd Dirac SS located 1.5 eV above the 1st CB edge, and we showed how excitation with 1.5 eV pho- tons drives a direct optical transition between these two states. To our knowledge, this is the first measurement on any material using a single photon energy for both 1PPE and 2PPE. This is achieved by using a photon energy exceed- ing the sample work function, which has conventionally been avoided in 2PPE measurements. Since the only ex- perimental parameter tuned between 1PPE and 2PPE modes of measurement is the intensity, this represents an exceptionally simple method to determine both the occupied and unoccupied band structure of a material. This has exciting implications, for example, for materi- als such as the high-Tc cuprates, where a measurement of both the occupied and unoccupied sides of the energy gap would provide insight on the particle-hole asymmetry of the superconducting and pseudogap states [45, 46]. The demonstrated ability to optically excite the 2nd SS has a number of interesting implications. The 2nd SS is expected to have all the novel topological and spin properties which characterize the well-known 1st Dirac SS of TIs [21, 33]. The results of Fig. 4 demonstrate that it is resonantly populated by 1.5 eV", + "optically excite the 2nd SS has a number of interesting implications. The 2nd SS is expected to have all the novel topological and spin properties which characterize the well-known 1st Dirac SS of TIs [21, 33]. The results of Fig. 4 demonstrate that it is resonantly populated by 1.5 eV photons. This could have relevance for the number of existing stud- ies on n-type TIs which have utilized 1.5 eV photons [11, 12, 14, 16, 17, 19, 47]. These experiments have been interpreted without knowledge of the unoccupied topological SS, and it would be interesting to evaluate whether the transition into this state plays a role in the physics discussed in those results. Finally, the fact that it can be accessed by 1.5 eV photons is particularly appro- priate for applications, since this is the fundamental pho- ton energy provided by commercial ultrafast Ti:Sapphire lasers. This discovery therefore demonstrates a unique opportunity for direct ultrafast optical coupling to TI SSs. We then proceeded to study the electron dynamics initiated by optical excitation of p-type Bi2Se3. We found that the dynamics culminate in a persistent non- equilibrium population of the spin-textured SS, which is attributed to continuous filling from a meta-stable popu- lation in the CB. We note that excitation with any above- bandgap photon energy should lead to the same SS filling behavior, since inter- and intra-band scattering processes inevitably lead to relaxation of carriers toward the CB edge. This phenomenon could find a role in applications requiring ultrafast optical control of a spin-polarized sur- face conduction channel. Finally, we focused on the relaxation of electrons in the CB of p-type Bi2Se3. We found an unexpected decrease of relaxation rates with increasing sample temperature, and showed that this behavior is reproduced by a sim- 9 ple electron-phonon scattering model. While we have demonstrated this for a nearly-empty band far from EF , we believe these considerations to be equally significant for dynamics near EF . In fact, it is straightforward to extend this model to this regime by simply removing the assumption that f ≪1. This is not within the scope of the current work, but believe it would be an interesting avenue for future study. This analysis serves as an important reminder that the population decay rate extracted by trARPES is not in general equal to the single electron scattering rate. Rather, the dynamics measured by trARPES is that of a distribution of electrons, which naturally includes a com- bination of filling and emptying processes from a con- tinuum of states. In general, the contributions from all co-existing processes must be carefully considered when interpreting transients from trARPES data. ACKNOWLEDGMENTS We thank M. Sentef for valuable discussions. This work is supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Science un- der contract DE-AC02-76SF00515. J. A. S. and S.-L. Y. acknowledge support by the Stanford Graduate Fellow- ship. P. S. K. acknowledges support by the Alexander- von-Humboldt foundation through a Feodor-Lynen fel- lowship and continuous support by M. Wolf. A. F. .K. is supported by the", + "Division of Materials Science un- der contract DE-AC02-76SF00515. J. A. S. and S.-L. Y. acknowledge support by the Stanford Graduate Fellow- ship. P. S. K. acknowledges support by the Alexander- von-Humboldt foundation through a Feodor-Lynen fel- lowship and continuous support by M. Wolf. A. 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[45] M. Hashimoto, R.-H. He, K. Tanaka, J.-P. Testaud, W. Meevasana, R. G. Moore, D. Lu, H. Yao, Y. Yoshida, H. Eisaki, T. P. Devereaux, Z. Hussain, and Z.-X. Shen, Nature Physics 6, 414 (2010). [46] B. Moritz, S. Johnston, T. P. Devereaux, B. Muschler, W. Prestel, R. Hackl, M. Lambacher, A. Erb, S. Komiya, and Y. Ando, Physical Review B 84, 235114 (2011). [47] J. W. McIver, D. Hsieh, H. Steinberg, P. Jarillo-Herrero, and N. Gedik, Nature nanotechnology 7, 96 (2012).", + "arXiv:1711.09828v1 [physics.app-ph] 27 Nov 2017 Electronic and magnetic properties of the graphene densely decorated with 3d metallic adatoms Ma lgorzata Wawrzyniak-Adamczewska Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´n, Poland E-mail: mwaw@amu.edu.pl Abstract. The electronic properties of graphene decorated with Ni, Co, Cu and Zn adatoms is studied with the density functional theory approach. Within the analysis the spin-orbit interaction is taken into account. We focus on the case when the indicated 3d metallic adatoms form a perfect, close-packed single-atomic layer above the graphene surface. The two configurations are examined, namely the adatoms in the on-top, and the hollow positions on graphene. First, we verify that the metallic adatoms in the close-packed structure do not form a covalent bonds with the graphene substrate. However, due to the proximity of the metallic adatoms to the graphene, the charge transfer from the adatom layer to the graphene takes place, and in consequence the graphene becomes n-doped. The observed charge transfer results from the arising hybridization between the graphene 2p and transition metal 3d orbitals. The proximity of metallic adatoms modifies the magnetic state of the graphene. This effect is especially pronounced for the decoration with magnetic atoms, when the magnetic moments on the graphene sublattices are induced. The analysis of the band structure demonstrates that the charge transfer, as well as the induced magnetism on graphene, modify the graphene electronic properties near high symmetry points, especially the Dirac cones. The presence of the metallic adatoms breaks graphene K −K ′ symmetry and splits the bands due to the exchange coupling. We show that for the hollow configuration the gap opening arises at the K(K ′)-point due to the Rashba-like spin- orbit interaction, while in the case of the on-top configuration the energy gap opens mainly due to the staggered potential. We also mapped the parameters of an effective Hamiltonian on the results obtained with the density functional theory approach. 1. Introduction Graphene is one of the most prominent currently studied two dimensional material [1]. This stable single layer of carbon atoms forms a honeycomb lattice and possesses extraordinary electronic properties, i.e. a linear band dispersion for low energies near K and K ′ symmetry points. The exciting electronic properties of graphene originate from the fact that its structure consists of the two equivalent sublattices, with the assigned pseudospins [2, 3, 4]. 2 Since it is technologically inconvenient to obtain free standing graphene and incorporate it into electronic devices, the graphene growth on various substrates, as well as its decoration with various adatoms, or formation of the hetero-layered structures addresses much attention [5]-[20]. The structural incorporation of graphene with other materials may lead to the enhancement of the spin-orbit interaction, which destroys the perfect massless-relativistic picture for charge carriers and introduces the gap in the energy spectrum as well as a spin-splitting of the bands. This feature is desirable, if addressing the graphene as a part of the electronic devices for spin-dependent transport, spin-filtering and magnetic valves. On the other hand, the unbounding of the graphene from substrates, e.g. by intercalation, in order", + "in the energy spectrum as well as a spin-splitting of the bands. This feature is desirable, if addressing the graphene as a part of the electronic devices for spin-dependent transport, spin-filtering and magnetic valves. On the other hand, the unbounding of the graphene from substrates, e.g. by intercalation, in order to recapture its unique relativistic properties is also desirable. The magnitude of the intrinsic spin-orbit coupling in pristine graphene is small – of orders of tens µeV – and originates from the graphene d-orbitals hybridization [27]. Therefore, a scientific effort is done to study the rˆole of substrate or decoration with adatoms. First theoretical and experimental works analyzed the rˆole of the metallic substrate on the electronic properties of graphene [21]-[26], however the relativistic effects were not taken into account in those studies. Recently, several theoretical works appeared considering quantitatively the influence of the spin-orbit interaction on the symmetry breaking near K, K ′ points and its influence on the modification of the linear dispersion of the Dirac cone for the graphene placed on the metallic surfaces [28, 29, 30, 31, 32]. The properties of grahene decorated with metallic adatoms was also discussed [33]. In this paper we focus on how the electronic properties of the graphene are modified when the graphene is decorated with a single layer of the close-packed 3d metallic adatoms. The paper is organized as follows. First, the considered structures and applied methods are described. Next, taking into account the spin-orbit interaction as implemented in the applied pseudopotential DFT method, the general electronic properties, regarding charge transfer and the modification of the band structure of the graphene in the proximity of the metallic layer are considered. Subsequently, the insight into the orbital hybridization between the graphene and the metallic layer is presented. Then, the proximity induced magnetism on graphene is described. In the end, the quantitative analysis of the influence of the spin-orbit interaction of the Rashba- like type, the exchange coupling and the staggered potential at the Dirac point of the specified heterostuctures is presented and discussed. 2. Method and System Geometry The presented results were obtained using a DFT approach as implemented in the plane-wave pseudopotential Quantum Espresso code [34]. We used fully- relativistic pseudopotentials in the Perdew-Burke-Ernzerhof (PBE) parametrization for the exchange–correlation functional [35]. A plane-wave energy cutoffwas set to 80 Ry, while charge density cutoffto 600 Ry, for all atomic species. The applied pseudopotntials 3 contain the 2s, 2p projections of valence states for carbon atoms, while 3d, 4s and 4p projections for metallic adatoms. Within the analysis the Hubbard U corrections for the 3d adatomic orbitals, in rotationally invariant scheme, were taken into account [36]. The value of the effective U parameter for the nickel was set to U = 6 eV, while for the cobalt and zinc we assumed U = 2 eV [37], and for copper we set U = 1 eV. We also took into account the semi–empirical van der Waals corrections, since they are important for the large systems with dispersion forces [38, 39]. The considered hexagonal supercells consists", + "for the cobalt and zinc we assumed U = 2 eV [37], and for copper we set U = 1 eV. We also took into account the semi–empirical van der Waals corrections, since they are important for the large systems with dispersion forces [38, 39]. The considered hexagonal supercells consists of the two graphene atoms, of A and B sublattices, and the metallic adatom in the on-top or hollow positions. The geometries of the considered structures with indicated supercellls are presented in Fig.1. The periodic slabs were separated by 12 ˚A of vacuum. We used a uniform 30×30×1 Monkhorst- Pack k-mesh [40] to sample the first Brillouin zone for the hexagonal supercells. Within the applied parameters and pseudopotentials, the optimized C–C distance of pristine graphene is 1.42 ˚A. Figure 1. The top view of the graphene decorated with adatoms in the hollow (left) and the on-top (middle) configurations. The specified adatoms (blue) form a close- packed layer above the graphene (yellow). The picture on the right hand side shows the side view of the two-layered structure in the on-top configuration. The graphene– metallic layer distance differs for adatom species and their positions on graphene, what is shown in Tab.1. Since we focus on the close-packed decoration, the choice of the specified adatoms is motivated by the adequate matching of the lattice constants of graphene and the lattice constants of the established metallic layers. We focus on the studies of decoration with nickel and copper atoms, since the Ni–Ni interactomic distance on the Ni(111) surface is 2.48 ˚A and the Cu–Cu interatomic distance on Cu(111) surface is 2.55 ˚A, as well as with cobalt and zinc atoms, since the interatomic distances for the Co(0001) and Zn(0001) close-packed surfaces are 2.51 ˚A and Zn–Zn 2.66 ˚A, respectively [41, 42]. The optimized distances between the graphene and the close-packed adatomic layers are given in Tab.1. The presented values suggest the physisorbtion of the 3d metallic adatoms on the graphene substrate. This statement is then confirmed by the analysis of the binding energy calculated per grahene C–atom. The values of the binding energy, 4 Table 1. The values of the optimized graphene – adatom layer distances d and in ˚A, for the on-top (t) and hollow (h) configurations. Ni(h) Ni(t) Co(h) Co(t) Cu(h) Cu(t) Zn(h) Zn(t) d 3.3754 3.0648 3.2112 3.1218 3.0787 3.1011 3.4636 3.3154 Table 2. The values of the total energies in Ry for the on-top and hollow configurations of the graphene decorated densely with the 3d metallic adatoms. Ni Co Cu Zn on-top -124.29904316 -99.88880617 -141.30003430 -172.23111898 hollow -124.29798240 -99.88733592 -141.30002475 -172.23130580 presented in Tab.3, are far lower than 500 meV, what allows us to conclude that no covalent bounds are formed between the graphene and the close-packed adatoms. The values of the total energy for the graphene with metallic adatoms is shown in Tab.2. From the analysis of the total energies in the specific configurations it is clearly seen, that the on-top configuration is the preferred one for nickel and cobalt layers, namely for the ferromagnetic metallic layers. The difference in total", + "total energy for the graphene with metallic adatoms is shown in Tab.2. From the analysis of the total energies in the specific configurations it is clearly seen, that the on-top configuration is the preferred one for nickel and cobalt layers, namely for the ferromagnetic metallic layers. The difference in total energy between the hollow and on-top configurations is of order of 14 and 20 meV for the nickel and cobalt layers, respectively. This suggests that, while close-packed, the mentioned atoms do not tend to nest the hollow position on the graphne lattice and rather be imbedded in the on-top position. The difference of the total energies for the copper layers is of order of few µeV merely, in favor of the on-top configuration, while the zinc atoms prefer to nest the hollow position, and the difference between the total energy in the on-top and hollow configurations for the latter adatom species is found to be equal 2.5 meV. At this point we emphasize however, that supercell we applied to our DFT analysis is too small to include the effects of the Moire patterns of the graphene–metal interface [43, 44, 45, 46], as well as the buckled graphene patterns [47]. Hence, the analysis of the spatially extended system, with larger supercells, could led us to different conclusions regarding the structure privileges. Moreover, we verified that the reduction by half of the adatoms concentration on the graphene significantly changes the bonding type. Namely, when the adatoms are spread on graphene at lower concentrations they tend to form covalent bonds with the graphene substarate. 3. Results 3.1. Band structure In order to verify the results with spin-orbit coupling, first we calculated the intrinsic spin-orbit induced gap for pristine graphene at the K-point. With the established DFT 5 Table 3. The values of the binding energies in eV for the on-top and hollow configurations of the graphene decorated densely with the 3d metallic adatoms. Ni Co Cu Zn on-top 133.34 107.66 62.82 103.94 hollow 117.45 87.74 62.69 106.49 parameters, supercells and pseudopotentials we obtained value of the intrinsic spin-orbit gap at 10 µeV . This value stays in excellent agreement with the previous DFT results [48] and also gives a precision of our spin-orbit induced energy gap calculations that are presented in the final part of this work. In Fig.2, the band structures for the graphene densely decorated with 3d metallic adatoms for the hollow and the on-top positions are shown. In the on-top case the metallic adatom is situated above the graphene carbon atom of the A sublattice, hence a coupling with the A sublattice is assumed to be stronger than with the B sublattice. For the hollow configuration the equal couplings with both sublattices are present. The adatom type and position on the graphene lattice is indicated in the figure. The band structure for the pristine graphene is plotted with the purple line and presented in all figures for compartion. The figures show the course of the band structures obtained in the calculations where the Hubbard U corrections of the 3d metallic orbitals are", + "lattice is indicated in the figure. The band structure for the pristine graphene is plotted with the purple line and presented in all figures for compartion. The figures show the course of the band structures obtained in the calculations where the Hubbard U corrections of the 3d metallic orbitals are or are not taken into account, as indicated. For the presented energy range, there is no much difference in the band structure between the hollow and the on-top configurations along reciprocal path for the particular considered 3d adatomic layer. The exception is the vicinity of the M-point, where the difference in the band structure for the conduction states between the hollow and the on-top cases is pronounced. This difference is especially pronounced for Ni, Co and Cu adatomic layers around 2 eV, and appears despite the treat of the Hubbard U corrections. The difference in the band structure course near the M-point is attributed to the fact that for the on-top configuration the symmetry breaking appears in the reciprocal space near M-point on M–Γ path. At the presented energy range, the linear dispersion near K symmetry point is reproduced for all considered adatoms. However, the shift in energy of the Dirac point with respect to the Fermi Energy EF, here denoted by ED, appears. We attribute this shift to the strength and type of the hybridization between the graphene and adatomic layer. The n-doping type is observed for all studied adatoms, since the Dirac point of the considered systems is shifted below the Fermi energy. It is worth to underline, that taking into account the Hubbard corrections in the case of nickel layer is crucial, since it alters the graphne doping type, while for the zinc adatomic layer the modification in the course of the band structure introduced by the Hubbard correction is unnoticeable in the presented energy range, despite the assumption of a quite large value of the U parameter. In general, the two groups of the adatoms among the studied ones may be distinguished. There are the adatoms that tend to hybridize strongly with the graphene, 6 Table 4. The values of shift of the Dirac point, ED in eV, for the indicated 3d adatoms in the hollow and on-top configurations. Ni Co Cu Zn on-top -0.9710 -0.5869 -0.1425 -0.9059 hollow -0.9824 -0.3329 -0.1718 -0.7170 and for them the shift of the Dirac point is significant – these are nickel, cobalt and zinc, and the adatoms that can be regarded as weakly coupled to the graphene – the copper. The values of the shift in energy of the Dirac point ED for all considered adatoms are presented in Tab.4. Our observation of the graphene doping type stays in agreement with the previous studies of the graphene placed on the metallic substrates [26, 49]. One may also observe, that the set of flat doping bands builds up below or/and around the Fermi energy. The position and spread in energy of the set of the doping bands depends on the adatom type. For the presented energy window the set of doping bands", + "substrates [26, 49]. One may also observe, that the set of flat doping bands builds up below or/and around the Fermi energy. The position and spread in energy of the set of the doping bands depends on the adatom type. For the presented energy window the set of doping bands is clearly visible for the nickel, cobalt and copper adatomic layers, while for the zinc the doping bands lay below the presented energy frame. Importantly, for the nickel and cobalt adatoms, and lesser for the copper layer, the location in energy of the doping bands depends on the strength of the Hubbard U corrections. In general, taking into account the Hubbard U corrections results in dragging down in energy of the valence bands, while the conduction states are less affected by these corrections. In the case of nickel, cobalt and to some extend copper, the increase of the U correlations enhances the shifting down the doping bands. As mentioned above, the inclusion of the Hubbard correction not only influences the position of the doping bands, but also the position in energy of the Dirac point. Here, the two types of the adatoms may be extracted. The adatoms for which the Hubbard correction notably shifts down the Dirac point and/or the position of the doping bands – for this group the adatoms forming the ferromagnetic layers, namely nickel and cobalt atoms are included. The other group consists of the adatoms, for which the Hubbard U corrections do not influence the position of the Dirac point and the doping bands are merely moved. The later group represents the copper and zinc, namely nonmagnetic adatoms. The effect responsible for the shifting down the band structure, and in particular, the Dirac point below the Fermi energy is the charge transfer that takes place from the metallic layer to the graphene. This charge transfer is possible since the hybridization between the specific graphene and adatomic layer orbitals develops. The n-type doping for graphene on metallic surfaces is predicted theoretically [28, 22] and reported in experiment [50]. 7 3.2. The p −d hybridization The analysis of the adatomic layer – graphene hybridization for the hollow and the on- top configurations is presented in Fig.3 and Fig.4, respectively. The left panel of these figures shows the band structure projected on the indicated atoms of the considered supercells. The provenance of the build-in flat doping bands, the modification of the graphene-based bands, as well as the crossings and anti-crossings between the adatomic and graphene states in the reciprocal space are visible. The middle panels of Fig.3 and Fig.4 show the band structures projected on the adatomic 3p and 3s, as well as the graphene 2s states, while the right panels show the projections on the adatomic 3d and graphene 2p states. In the case of the nickel, copper and zinc layers the grapehne 2s and adatomic 3s, 3p and 3d states do not constitute the graphene cone. For the mentioned layers the graphene cone is constituted by the carbon 2p states only and the contribution from the states of the", + "states. In the case of the nickel, copper and zinc layers the grapehne 2s and adatomic 3s, 3p and 3d states do not constitute the graphene cone. For the mentioned layers the graphene cone is constituted by the carbon 2p states only and the contribution from the states of the A and B sublattices is equal for the hollow configuration, while in the on-top configuration the modest difference for the states of the A and B sublattives is observed. The exeption is the cobalt layer. For the cobalt in the on-top confguration the p −d hybridization encompasses the bands forming the Dirac cone. The flat doping bands originate from the adatomic 3d states. For the cobalt layer these states are also found at the energy range where the Dirac point is located, and for the conduction states. The hilly bands crossing the Fermi energy, that originate from the adatomic 3p states for zinc, and from adatomic 3p and 3d states for nickel, cobalt and carbon atoms, are also visible. Hence, the in-plane electron transport across the considered hybrid structure involves not only the graphene π-states, or in the case of the cobalt layer – the low velocity doping states, but also the 3p states for zinc layer and 3p plus 3d states of the nickel, cobalt and copper metallic layers. This analysis provides us the information for the proper construction of an effective Hamiltonian in the charge carrier transport description, where not only the graphene 2p and adatomic (or substrate) 3d states should be taken into account, but also the 3p doping states should be included. 3.3. The magnetism induced in graphene due to the proximity of an adatomic metallic layer Due to the proximity of the adatomic layers the finite values of the magnetic moments are induced on the graphene sublattices. The induced magnetism and its impact on the band structure depends significantly on the adatom type (whether they are magnetic or not) and the configuration on the graphene lattice. Although the total magnetic moment of the considered supercells containing nonmagnetic adatoms is zero, we find the small values of the induced magnetic moments on the graphene as well as on the metallic adatoms, for the studied nonmagnetic metallic layers. Hence, the finite values of the exchange coupling should be taken into account while describing the system with an effective Hamiltonian. The analysis of the proposed Hamiltonian is discussed in the next part of this paper. We also notice, that the tiny induced magnetic moments on the adatoms and the graphene sublattices are oriented 8 Table 5. The values of magnetic moments on the specified adatoms and induced on the carbon atoms of the graphene A and B sublattices, in the µB, for the hollow and on-top configurations. In the case of the on-top configuration the adatoms are located above the A sublatice. Ni CA (Ni) CB (Ni) Co CA(Co) CB(Co) on-top 1.0521 -0.0031 0.0012 1.8458 -0.0069 0.0067 hollow 1.0503 0.0045 0.0045 1.9232 -0.0008 -0.0008 perpendicularly to the system surface and oriented in line. For the ferromagnetic layers we observe that", + "of the on-top configuration the adatoms are located above the A sublatice. Ni CA (Ni) CB (Ni) Co CA(Co) CB(Co) on-top 1.0521 -0.0031 0.0012 1.8458 -0.0069 0.0067 hollow 1.0503 0.0045 0.0045 1.9232 -0.0008 -0.0008 perpendicularly to the system surface and oriented in line. For the ferromagnetic layers we observe that for the on-top, as well as for the hollow configurations, magnetic moments located on the adatoms and those induced on graphene, are oriented perpendicularly to the surface of the two-layered structure. However, the aligment of the magnetic moments differs for on-top and hollow configurations. In the on-top configuration, the alignment of the magnetic moments of the carbon atoms located below the adatom (let us assume it is the graphene A sublattice) is opposite to the aligment of the magnetic moments on the adatoms, while the magnetic moments induced on the other graphene sublattice – B, are aligned in the same direction as the moments on the adatoms. In the hollow configuration the aligment of the induced magnetic moments on the both graphene sublattices is opposite (cobalt) or consistent (nickel) to the direction of the magnetic moments localized on the adatoms. Hence, in the on-top configuration, the proximity of the adatomic layer causes the anti-ferromagnetic aliment on the graphene sublattices, while in the hollow configuration the ferromagnetic alignment on the graphene sublattices is forced by the proximity of the localized magnetic moment, that equally couples to both graphene sublattices. The values of the magnetic moments located on the specified magnetic atoms for the studied structures are presented in Tab.5. The proximity of all studied adatomic layer causes the exchange splitting of the graphene bands. This splitting is pronouced for the ferromagnetic adatomic layers and influences the electronic properties of the system. The exchange coupling and resulting magnetic splitting effect is particularly important in analyzing the band structure landscape near the Dirac point. 3.4. Energy dispersion in the vicinity of the Dirac point In the case of the proximity of the metallic layer to the graphene, the electronic properties near the Dirac point can be mapped on the following effective Hamiltonian [31, 32] H = H0 + H∆+ Hexch + HR, (1) where H0 = ¯hvD(τkxσx + kyσy) describes the kinetic energy of the charge carriers, with kx and ky dentoting the components of the charge carrier wave function, and σx, σy 9 Table 6. The values of indicated parameters of the Hamiltonian given by Eq.1 in meV, for the studied adatomic layers in the hollow (h) and on-top (t) configurations. ∆ λA λB λR Cu(t) 7.885 1.973 0.159 1.6 Cu(h) – 1.829 -1.849 1.8 Zn(t) 19.663 1.726 1.367 1.5 Zn(h) – 0.228 -0.208 0.2 Ni(t) 29.534 60.820 8.907 4.9 Ni(h) – -49.287 50.460 4.6 Co(t) 53.751 72.654 4.732 5.3 Co(h) – 67.853 -71.224 5.0 denote the Pauli matrices, while vD stands for the electron velocity at the Dirac point and τ = ±1 allows to distinguish between the K(K ′) points. This term refers to the gapless Dirac states near K(K ′) points. The proximity of the metallic layer to the graphene results", + "5.0 denote the Pauli matrices, while vD stands for the electron velocity at the Dirac point and τ = ±1 allows to distinguish between the K(K ′) points. This term refers to the gapless Dirac states near K(K ′) points. The proximity of the metallic layer to the graphene results in symmetry beaking of the pseudospins attributed to the graphene sublattices and the formation of the staggered potential that is felt by the pseudospins. This symmetry beaking results in formation an energy gap of the width 2∆at the Dirac point, and can be described with the Hamiltonian H∆= ∆σzs0, where the s0, denotes the unit matrix in the spin-space, while σz denotes the Pauli matrix. The proximity exchange effects are then described by the Hamiltonian [31] Hexch = λA[(σz + σ0)/2]sz + λB[(σz −σ0)/2]sz, (2) with the λA and λB being the exchange coupling constants (in the sense of the magnetic exchange coupling constants) for the indicated graphene sublattice. We notice, that in the hollow configuration, for the adatoms equally coupled to both graphene sublattices λA ≈−λB. For the on-top configuration, when the coupling to one of the graphene sublattices (let us assume A) is much stronger in comparsion to the coupling with the other subllatice, λA >> λB. Finaly, the proximity effect may lead to the enhancement of the Rashba-like interaction on the graphene, hence one should also take into account the term [32] HR = λR(τσxsy −σysx), (3) The mapping of the described effective Hamiltonian on the DFT analysis, given in allows us to extract the parameters, that are gathered in Tab.6. The band structures near the Dirac point for the graphene in the proximity of the copper and zinc metallic layers are presented in Fig.3.4. The general feature can be seen, namely, due to the staggered potential in the on-top configuration the energy gap at the K(K ′) is formed for all studied adatomic layers and this energy gap is additionaly widen by the Rashba-like interaction. On the other hand, for the hollow configuration the energy gap at the K(K ′) results from the spin-orbit coupling only. This is clearly 10 seen when comparing the band structures at the K point for the case when the spin- orbit interaction is included in the DFT calculation, with the case when calculations are performed assuming only spin-polarized system. 4. Conclusions Within this work we studied the electronic and magnetic properties of the two-layered structure, namely when the graphene is decorated densely with the 3d metallic adatoms. We found that no covalent bonds are formed between the graphene carbon atoms and the metallic adatoms for the perfect dense decoration. The n-type doping of the graphene is found for all studied cases and is caused by the charge transfer from the metallic layer to the graphene. This charge transfer involves the orbital hybridization of the 2p graphene states and 3d as well as 3p states of the adatomic layer. The two types of the adatomic layers may be distinguished among the studied layers, namely the adatoms that interact strongly with the graphene (nickel,", + "to the graphene. This charge transfer involves the orbital hybridization of the 2p graphene states and 3d as well as 3p states of the adatomic layer. The two types of the adatomic layers may be distinguished among the studied layers, namely the adatoms that interact strongly with the graphene (nickel, cobalt and zinc) and those which interact weakly – copper. The proximity of the metallic layer leads to the modification of the graphene electronic properties not only via the graphene charge doping but also through the exchange coupling arising between the graphene and metallic layers, the shape of the staggered potential and the spin-orbit Rashba-like interaction. The exchange coupling is pronauced for the ferromagnetic layers, however we find the small finite values of the exchange coupling also for the nonmagnetic layers. We show that the proximity effect visibly modifies the Dirac cone and this modification depends on the adatomic configuration, namely if the adatom is coupled equally to both graphene sublattices (hollow) or the coupling with one of the sublattices dominates (on-top). In the case of the on-top as well as hollow configurations the band spin-splitting arises due to the exchange coupling of the graphene and the metallic layer. This spin-splitting is of the the order of magnitude larger for the ferromagnetic layers than for the non-magnetic layers. Additionally, for the on-top configuration the energy gap apprears, due to the staggered potential. This energy gap is then modified by the spin-orbit Rashba-like interaction, although this modification is insignificant in comparsion to the value of the gap. On the other hand, for the hollow configuration, that Rashba-like interaction introduces the energy gap at the Dirac point, since the contribution form the staggered potential is absent. 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Klesse, A.Becker, F. Spth, Ch. Papp, E. Voloshina, Decoupling of graphene from Ni(111) via oxygen intercalation, Carbon Volume 121, September 2017, Pages 10-16, DOI: 10.1016/j.carbon.2017.05.068 14 Figure 2. The band structures for the graphene with close-packed metallic layers. The hollow and on-top configurations are specified in the figures. The band structures were calculated for the GGA only, or with additional assumption of the finite value of the Hubbard U parameter, as specified in the figure. The band structure for the pristine graphene is plotted with the purple line for compartion. 15 Figure 3. The band structures of the graphene densely decorated with the 3d metallic adatoms and projected on the indicated type of atoms (left panel). The middle panel presents the projection on adatomic 3s and 3p orbitals, as well as graphene 2s states. The right hand side panel presents the projection on the adatom 3d states and graphene 2p states. In the presented case the adatoms are placed in the hollow positions above the graphne. The Hubbard corrections are taken into account within the calculations. 16 Figure 4. The band structures of the graphene densely decorated with the 3d metallic adatoms in the on-top positions above the graphne, projected on the indicated atoms and states as in Fig.3. 17 Figure 5. The band structure in", + "The Hubbard corrections are taken into account within the calculations. 16 Figure 4. The band structures of the graphene densely decorated with the 3d metallic adatoms in the on-top positions above the graphne, projected on the indicated atoms and states as in Fig.3. 17 Figure 5. The band structure in the vicinity of the Dirac point for the on-top and hollow configurations for all considered adatomic metallic layers. The band structure for the spin-polarized case, when no spin-orbit coupling is taken in to account is also shown for compartion.", + "Estimated length: 8 pages TITLE Effect of discrete impurities on electron transport in ultra-short MOSFET using 3D Monte Carlo simulation AUTHORS Philippe Dollfus, Arnaud Bournel, Sylvie Galdin, Sylvain Barraud(*), and Patrice Hesto Institut d'Electronique Fondamentale UMR 8622 CNRS / Université Paris-Sud XI 91405 Orsay cedex, France e-mail: philippe.dollfus@ief.u-psud.fr (*) present address: LETI/DTS, CEA/Grenoble, 17 rue des martyrs 38054 Grenoble cedex 9 ABSTRACT This paper discusses the influence of the channel impurity distribution on the transport and the drive current in short-gate MOSFET. In this purpose, a careful description of electron-ion interaction suitable for the case of discrete impurities has been implemented in a 3D particle Monte Carlo simulator. This transport model is applied to the investigation of 50 nm MOSFET operation. The results show that a small change in the number of doping impurities or in the position of a single discrete impurity in the inversion layer may significantly influence the drain current. This effect is not only related to threshold voltage fluctuations but also to variations in transport properties in the inversion layer, especially at high drain voltage. The results are analyzed in terms of local fluctuations of electron velocity and current density. In a set of fifteen simulated devices the drive current Ion, determined at VGS = VDS = 0.6 V, is found to vary in a range of 23% according to the position of channel impurities. KEYWORDS MOSFETs, semiconductor device modeling, Monte Carlo methods, semiconductor device doping, doping fluctuations. 2 I. INTRODUCTION In many respects related to physical and technological limitations the downscaling of MOSFETs to sub-50 nm dimensions poses in new terms the design of next CMOS generations. Among the new important issues, the inescapable statistical fluctuations in the number and position of doping impurities yield spreading in device characteristics as the threshold voltage (VT) and the off-state current, to such a point that it may be unacceptable to CMOS operation. This limitation was early predicted by Keyes in 1975 [1] and experimentally demonstrated in the mid-1990s [2-4]. The statistical VT-fluctuations induced by dopant random distribution have been deeply investigated by analytical approaches [3,5] and by numerical drift-diffusion (DD) simulations using either continuous doping [5,6] or 3D atomistic models [7-9]. Even though the DD method is very effective in providing a large amount of results for statistical studies, 3D Monte Carlo (MC) simulations should give a more realistic description of transport, specially in the case of ultra-small device. Moreover, in most of atomistic simulations the discrete nature of impurities is only included in the charge assignment to the grid cells. The description of the long-range part of electron-ion interaction is then accounted for from the solution of the 3D Poisson equation that gives the so-called mesh force acting on the carriers. But it was shown that the short-range part of the interaction must be carefully taken into account and distinguished from the long-range part to correctly describe the Coulomb force [10-15]. 3D MOSFET simulations using a coupled Monte Carlo/molecular dynamics algorithm to properly include the short-range effects were recently reported [16]. In the present article we use a", + "part of the interaction must be carefully taken into account and distinguished from the long-range part to correctly describe the Coulomb force [10-15]. 3D MOSFET simulations using a coupled Monte Carlo/molecular dynamics algorithm to properly include the short-range effects were recently reported [16]. In the present article we use a pure Monte Carlo approach which includes the short-range part of the Coulomb interaction via a specific scattering mechanism applied in predefined short-range zones (SRZ). The detailed description of this electron-ion interaction model for 3D particle simulation may be found in [15]. However, some of its most important features are given in the next section to make clear the rest of the article. The model is validated through the computation of electron drift mobility in both N-type and P-type 3D Si resistors and the comparison with experimental results. Sec. III is devoted to the results of 50-nm MOSFET's simulation. We analyze the influence of discrete impurities on the carrier transport in the channel, and the effect of typical small changes in the impurity distribution on the drain current characteristics. This work does not aim at giving a full statistical analysis of VT fluctuations, but at physically describing the influence of dopant number and position on internal quantities (field, velocity, density) and on 3 the drive current Ion. II. THE ELECTRON-ION SCATTERING MODEL This section is dedicated to the description of the electron-ion interaction model developed for the case of discrete impurities. It has been shown that the electrostatic field resulting from the solution of 3D Poisson equation cannot correctly describe the interaction in the vicinity of impurities [10-13]. The necessary correction usually consists in defining around each impurity a short-range zone (SRZ) in which the long-range Coulomb interaction is removed and replaced by a short-range mechanism. It has been proposed to couple Monte Carlo (MC) and Molecular Dynamics (MD) techniques to compute the short-range forces between particles inside the SRZs [10,11,13] using the so-called P3M algorithm [14]. In our MC model the short-range interaction is described by an instantaneous scattering mechanism characterized by a scattering rate [15] as for all other scatterings experienced by carriers. This approach is valid if we take care to consider three questions simultaneously: the choice of the mesh size, the choice of the SRZ size and the model of scattering rate. The choice of the mesh size is dictated by two considerations. First, one has to save the number of nodes on which is solved the 3D Poisson equation. Second, the mesh size ∆X must be small enough to correctly describe the smallest space fluctuations of potential to be considered in the simulation. Assuming an equilibrium region with an average concentration of discrete impurities N, the smallest space effect to be considered is the variation of Coulomb potential between impurities in order to calculate correctly the long-range carrier-ion interaction. A mesh size of one tenth of the average distance between impurities, i.e. 10 3 / 1 − = ∆ N X , is proved to be a good compromise [15]. In the case of majority carriers, i.e. attractive", + "between impurities in order to calculate correctly the long-range carrier-ion interaction. A mesh size of one tenth of the average distance between impurities, i.e. 10 3 / 1 − = ∆ N X , is proved to be a good compromise [15]. In the case of majority carriers, i.e. attractive impurities, the size of the SRZ must be large enough to avoid inconsistencies between a classical description of particles and a thin impurity potential that looks like a quantum box [15]. But to simplify the calculation of electron-ion scattering rate and to correctly describe the long-range interaction, the SRZ should be as small as possible. Indeed, if the volume of the SRZ is chosen smaller than the volume of the screening sphere one can neglect screening effect in the calculation of scattering rate. Given the choice of mesh spacing ∆X the good trade-off between these contradictory criteria is to extend the SRZ to two cells around impurities. For instance, if the carrier and impurity concentration is 1018 cm-3, the cells are cubes of side ∆X = 1 nm and the 4 SRZs are cubes of side 5 nm, i.e. containing 125 cells, centered on the impurity cells. It should be mentioned that the choice of the SRZ size is less critical in the case of repulsive impurities (minority carriers) than in the case of attractive impurities (majority carriers). Since the short-range scattering applies within a volume smaller than the screening sphere, i.e. the sphere of radius equal to the Debye length, we do not have to consider screening effects in the derivation of the scattering rate imp τ 1 [15]. Practically, we calculate imp τ 1 using the Takimoto approach [17] as for the case of continuous doping, with a doping concentration equal to 1 over the volume of SRZ. This model includes the screening effect but we assume a very long screening length µm 4.0 0 = L , which makes the screening influence on the scattering rate negligible. We have checked that by using a standard scattering rate with a screening length deduced from the local carrier density, the calculated mobilities are strongly overestimated. To validate the model, the electron drift mobility in both N-type and P-type silicon has been computed for comparison with experimental bulk-Si mobilities. The simulated structures are 3D resistors with two terminal contacts. The concentration of discrete doping impurities is in the range 1015-1019 cm-3. The number of simulated carriers is adjusted to obtain electrical neutrality in the structures. It should be noted that the number of particles is much higher than the number of ions (typically 2000 times higher), which means that we consider sub-particles [18]. It allows us to reduce the particle noise by statistical enhancement of rare events and to get an average number of particles by cell greater than one, which is crucial to stabilize the solution of Poisson's equation in quasi-equilibrium regime. The 3D Poisson's equation is solved at each time step varying in the range 0.1-1 fs, according to the carrier concentration, i.e. the doping concentration. By applying periodic boundary", + "number of particles by cell greater than one, which is crucial to stabilize the solution of Poisson's equation in quasi-equilibrium regime. The 3D Poisson's equation is solved at each time step varying in the range 0.1-1 fs, according to the carrier concentration, i.e. the doping concentration. By applying periodic boundary conditions, i.e. each carrier exiting through a contact is re-injected at the other side with the same wave vector, the number of carriers is constant. To compute the majority electron mobility, N-type resistors are simulated including electrons only. To compute the minority electron mobility we simulate P-type resistors with a given small amount of electrons: if the number of holes required to balance the ion charges is 0 P we simulate 1.1× 0 P holes and 0.1× 0 P electrons to maintain the neutrality. The resulting electron mobility is plotted in Fig. 1 (symbols) as a function of doping concentration and compared to experimental results given in [19] (N-type Si, continuous line) and [20] (P-type Si, dashed line). A very good agreement is found between calculated and 5 experimental mobilities in both cases of majority and minority electrons. It should be noted that this atomistic transport model makes very simply the difference between attractive and repulsive impurities. This difference only results from the 3D solution of Poisson equation, i.e. from the long-range part of the electron-ion interaction. III. MOSFET SIMULATION The atomistic model of impurity interaction has been applied to study the effect of the discrete character of channel dopants in short-gate MOSFET. Our intention is not to perform a full statistical analysis of dopant fluctuations, but to evaluate the influence of some specific impurity positions in the inversion layer on electron transport and device characteristics. The basic structure of simulated devices is shown in Fig. 2. The channel length L and width W are 50 nm. The junction depth ZJ is 15 nm and the gate oxide thickness tox equals 1 nm. To generate the reference device the doping implantation steps have been simulated using a Monte Carlo model [21]. In N-type source and drain regions the simulated average distribution of arsenic is approximated by a continuous doping profile decreasing from 5×1019cm-3 at the surface to 1018cm-3 at the junction depth. The distribution of dopants is considered as a discrete distribution only in the channel, i.e. in the so-called \"active region\" of volume L×W×ZJ. The boron implantation steps were optimized to give a uniform average concentration of 1018cm-3 on a large depth. It yields about 37 discrete impurities in the active region. At depth higher than ZJ the boron distribution is considered as a continuous concentration approximated by a smooth function. This article deals with the effect of small changes in the distribution of some of the 37 discrete impurities within the active region and more precisely in the inversion layer, which led us to study fourteen additional devices to be compared to the reference. Unless otherwise indicated, all results shown below have been obtained for a drain voltage VDS of 0.6 V. For a doping concentration of 1018cm-3 the average", + "the active region and more precisely in the inversion layer, which led us to study fourteen additional devices to be compared to the reference. Unless otherwise indicated, all results shown below have been obtained for a drain voltage VDS of 0.6 V. For a doping concentration of 1018cm-3 the average distance between impurities is 10 nm. According to the criterion defined above the mesh spacing ∆X is thus chosen equal to 1 nm. Around each impurity the SRZ is then a cube of side 5 nm. In principle, the size of each SRZ should depend on the local carrier density which may strongly change according to the applied bias and the impurity position. However, if the SRZ size is critical in the case of attractive impurities, it is much less in the case of repulsive ions as in a MOSFET channel. We have thus considered the same constant size for all SRZs. 6 The 3D Poisson's equation is solved at each time step equal to 0.1 fs with standard boundary conditions [22]. The number of simulated particles, which are still sub-particles, is typically 90000. In addition to electron-impurity interaction, the scattering mechanisms included in the Monte Carlo algorithm are phonon scattering and surface roughness scattering. The acoustic intra-valley phonon scattering is treated as an elastic process and the intervalley phonon transitions, consisting of three f-type and three g-type processes, are considered via either zeroth-order or first-order transition matrix [23] in agreement with selection rules. The phonon coupling constants given in [22] are used. Surface roughness scattering is treated with an empirical combination of diffusive and specular reflection which correctly reproduces the experimental universal mobility curve [24-25]. A fraction of 14 % of diffusive reflections has been considered in this work. A. Reference device As an introduction to the effects of discrete doping we start with some results obtained for the reference device. The cartography (top view) of the longitudinal current density Jx in the first sheet of cells under the gate oxide is shown in Fig. 3 for VGS = 0.45 V and VDS = 0.6 V. The presence of three impurities in the inversion layer is indicated by white circles and is characterized by low-current regions around impurities (dark) and high-current regions between them (bright). To analyze the transport in the channel we define in Fig. 3 two typical lines between source and drain. At Y = 38.5 nm the solid line does not go through any impurity and at Y = 8.5 nm the dashed line goes through one impurity (at XI = 23.5 nm). Along these two lines the profiles of longitudinal electric field (Ex component), electron density and electron velocity (vx component) are plotted in Figs. 4a, 4b and 4c, respectively. On the electric field profile plotted in dashed line, the presence of an impurity is characterized by a plateau in the 5 nm-wide SRZ, resulting from the vanishing of the mesh force, and by a repulsive field observed on both sides of the SRZ, i.e. positive on the left and negative on the right. The presence of", + "dashed line, the presence of an impurity is characterized by a plateau in the 5 nm-wide SRZ, resulting from the vanishing of the mesh force, and by a repulsive field observed on both sides of the SRZ, i.e. positive on the left and negative on the right. The presence of the impurity strongly affects the local density by generating an electron accumulation on the left side and a depletion on the right side (Fig. 4b). Moreover electrons considerably slow down around the impurity (Fig. 4c), which explains the weak current density observed locally in the cartography (Fig. 3). The largest part of the current circulates in the regions free of impurity where the electron velocity is higher. Seeing these strong effects of the presence of an impurity on the local quantities, a 7 significant influence of the number and the position of discrete dopants in the inversion layer can be expected. The rest of the article tackles this problem. B. Effect of the position along the channel of a single impurity In this sub-section, we analyze the effect of moving one of the discrete impurity in the active region along the source-drain X axis. We consider the impurity, visible in Fig. 3, whose initial position is XI = 9.5 nm, YI = 26.5 nm. Its position XI is moved from 2.5 nm to 47.5 nm. We first compare two cases: XI = 9.5 nm (initial case, i.e. reference device) and XI = 43.5 nm. The cartography of current density is shown in Fig. 5 for XI = 43.5 nm, i.e. in the case where the impurity is near the drain-end of the channel. It is thus in the high-field region (as shown in dotted line in Fig. 6a) and when compared to the previous case of Fig. 3 (XI = 9.5 nm) the current density is now higher and more homogeneous in the source-end region where electrons are injected (Fig. 5). The electric field and velocity profiles are plotted in Figs. 6 in three cases: along the line Y = 38.5 nm without any impurity (solid line, as in Fig. 4b), along the line YI = 26.5 nm with the impurity in XI = 9.5 nm (dashed line) and along the line YI = 26.5 nm with the impurity in XI = 43.5 nm (dotted line). If the impurity is in the source-end of the channel (dashed line) the injection-velocity is strongly degraded when compared to the case without impurity, but the overshoot velocity at the drain-end is similar in both cases (Fig. 6b). On the contrary, if the impurity is in the drain-end (dotted line) the maximum velocity significantly decreases but the injection-velocity is not affected when compared to the solid line. The consequence on the drain current is shown in Fig. 7a where ID is plotted as a function of the gate voltage VGS (at VDS = 0.6 V) in both cases. The current is about 3.8% higher for XI = 43.5 nm (dashed line) than for XI = 9.5 nm (solid line). The degradation of the injection velocity at", + "Fig. 7a where ID is plotted as a function of the gate voltage VGS (at VDS = 0.6 V) in both cases. The current is about 3.8% higher for XI = 43.5 nm (dashed line) than for XI = 9.5 nm (solid line). The degradation of the injection velocity at the source-end is thus more detrimental than the reduction of overshoot velocity at the drain-end, which is in agreement with other theoretical investigations [26-27]. Additionally, it should be noted that the difference between both curves is not a shift but rather a change in the slope, i.e. in the transconductance. In other words, changing the X position of an impurity induces a change in the channel transport properties which reflects on the drive current but probably not on the threshold voltage VT. It is confirmed by the ID-VDS characteristics plotted in Fig. 7b. At low VDS, i.e. under the usual conditions of VT 8 measurement, the current results from the average transport properties all along the effective channel length and is thus very weakly influenced by the X-position of a single impurity. Differences between these two devices only occur at high VDS, that is when the current is strongly dependent on the injection velocity at the source-end and thus influenced by the impurity position. The drive current Ion, defined at VGS = VDS = 0.6 V, is plotted in Fig. 8 as a function of the impurity position XI (solid line linking the full circles). The general trend to the increase of Ion as the position XI goes to the drain is confirmed. However, the current slightly increases also when the impurity goes very close to the source-end. In this case, indeed, the SRZ associated with the impurity and the repulsive field on the left side partially extend in the highly-doped source region. The influence of this impurity on the electron transport is then smaller. To be fully effective the ion must be plainly inside the active region, i.e. at XI ≥ 10 nm. C. Effect of an added or removed single impurity We now analyze the effect of changing the number Nimp of impurities in the inversion layer by one unit. There are 37 impurities in the active region of the reference device. We already mentioned the effect of the impurity located at position XI = 23.5 nm and YI = 8.5 nm (sub-section III.A), i.e. at a X position for which the effect of the impurity on the total current is maximum, as just shown (in Fig. 8). We consider two other devices by removing this impurity, yielding Nimp = 36, or by adding an impurity at the same X position but a different Y position (YI = 38.5 nm), yielding Nimp = 38. The cartography of current density in these two devices is shown in Figs. 9a and 9b, respectively, to be compared with Fig. 3. As expected, the homogeneity of the current density in the source-end region is obviously strongly dependent on these configurations, which should have significant effects on the total drain current. It is shown in", + "these two devices is shown in Figs. 9a and 9b, respectively, to be compared with Fig. 3. As expected, the homogeneity of the current density in the source-end region is obviously strongly dependent on these configurations, which should have significant effects on the total drain current. It is shown in Fig. 10 where ID is plotted as a function of VGS for the three devices. In this case the curves are almost parallel: the most important effect is thus a VT shift. Let come back to the Fig. 8 where the Ion value of the two new devices is plotted for comparison (open diamond and open square). The influence of the presence of a single impurity on Ion appears to be about 3.3%. Of course, in the case Nimp = 36, the effect on the current depends on the initial position of the removed impurity because of the variable electrostatic influence of other surrounding 9 impurities. As shown in Fig. 8 for this specific simulated device, removing the impurity located at XI = 23.5 nm (open diamond) yields a slightly smaller Ion value than removing the impurity located at XI = 9.5 nm (closed diamond). It should be noted that removing the impurity or placing it in the drain-end region induce very similar effects resulting in very close values of Ion. The effect of trapping of a single electron has been recently studied using atomistic DD simulation [28]. It is interesting to remark that relative current fluctuations of 5 % have been observed in strong inversion (at low drain bias), which is quite comparable with the effect of a single impurity obtained in the present work (at high drain bias). D. Effect of the vertical position of some impurities We now consider the effect of moving impurities vertically. Indeed, at a given number of impurities in the full active region (Nimp = 37 in our case), changing the number of impurities in the inversion layer is expected to modify the overall transport properties in the channel and the device performance. Starting from the reference device, another transistor has been generated by burying the three impurities visible just under the gate oxide (Fig. 3) 3.5 nm below the surface. This gives the current density mapping shown in Fig. 11a. The current density is then much more homogeneous in the inversion layer, even if buried impurities still have a small influence. On the contrary, the cartography shown in Fig. 11b corresponds to a device where four buried impurities have been moved to the surface, which yields the presence of seven impurities in the inversion layer. Two of these new surface impurities are close to the source-end and the two other ones are near the drain. Thus, as shown in sub-section III.B, they are not in a position where they can have the maximum influence on the transport. To enhance the effect of these four impurities they have been distributed along a line XI = 11.5 nm, which results in the cartography displayed in Fig. 11c. The regions of high current density are now very", + "in a position where they can have the maximum influence on the transport. To enhance the effect of these four impurities they have been distributed along a line XI = 11.5 nm, which results in the cartography displayed in Fig. 11c. The regions of high current density are now very limited when compared to the previous distribution shown in Fig. 11b and a fortiori to that of the reference device (Fig. 3). The consequence on the current characteristics and the value of Ion appears clearly in Fig. 12. Compared with the reference device (solid line), at VDS = 0.6 V and VGS = 0.45 V the current increases by about 9% by burying the three surface impurities (squares, dotted line) and decreases in the worst case by 22% with seven surface impurities (diamonds, dashed line). 10 At VGS = 0.6 V these figures become 5.8% and 16.8%, respectively. These current changes seem to be the consequence of both transconductance and threshold voltage fluctuations. E. Summary - Current statistics The number of simulated devices is certainly too small to make an accurate statistics on the effects of random fluctuations of doping impurity distribution in the inversion layer. However, we plot in Fig. 13 the bar chart of Ion values obtained for the fifteen devices, which summarizes the results obtained. This figure includes the result obtained by reversing source and drain in the reference device which increases Ion by 2.9%. In this set of devices the average Ion value is on I = 30.9 µA. The maximum and minimum Ion values are 32.7 µA and 25.6 µA, respectively. The simple effect of changing the position of a few impurities or adding/removing a single impurity in the inversion layer may thus induce drive current fluctuations in a range of at least 23% for this 50 nm MOSFET architecture. If we eliminate the worst case which is unlikely (Ion = 25.6 µA, corresponding to the distribution of Fig. 11c with six impurities almost aligned in the source-end region) the range of Ion fluctuations is still 11.2%. The relevant figure is probably in between these two values and thus is not negligible. We would like to point out again that these random current fluctuations observed at high drain voltage are not only due to threshold voltage shift but also in some cases to changes in channel transport properties reflected on the transconductance. A possible extension of this work could be the investigation of donor fluctuations, especially in the source region. IV. CONCLUSION Within 3D Monte Carlo simulation we have developed an electron-ion interaction model suitable for investigating the effect of dopant random fluctuations. The effect of fifteen typical distributions of channel doping atoms has been analyzed for a 50 nm MOSFET in terms of local physical quantities (as electron density and velocity, current density) and drain current. Small variations in the number and the position of these atoms are shown to significantly influence the transport properties in the inversion layer, which results not only in threshold voltage fluctuations but also, at high drain voltage, in transconductance variations.", + "(as electron density and velocity, current density) and drain current. Small variations in the number and the position of these atoms are shown to significantly influence the transport properties in the inversion layer, which results not only in threshold voltage fluctuations but also, at high drain voltage, in transconductance variations. As a consequence, the values of drive current are spread out on a range higher than 11% and 11 likely to reach more than 20%. It should be still larger in more aggressively downscaled devices, which probably reinforces the interest in investigating MOSFET architectures with undoped thin-channel for future CMOS generations. 12 13 REFERENCES [1] R. W. Keyes, \"The effect of randomness in the distribution of impurity atoms on FET thresholds\", Appl. Phys. vol. 8, pp. 251-259, 1975. [2] R. Sitte, S. Dimitrijev, and H. B. Harrison, \"Device parameter changes caused by manufacturing fluctuations of deep submicron MOSFET's\", IEEE Trans. Electron Devices, vol. ED-41, pp. 2210-2215, 1994. [3] T. Mizuno and J. Okamura, \"Experimental study of threshold voltage fluctuations due to statistical variation of channel dopant number in MOSFET's\", IEEE Trans. Electron Devices, vol. ED-41, pp. 2216-2221, 1994. [4] T. Mizuno, \"Influence of statistical spatial-nonuniformity of dopant atoms on threshold voltage in a system of many MOSFETs\", Jpn. J. Appl. Phys. vol. 35, pp. 842-848, 1996. [5] P. A. Stolk, F. P. Widdershoven, and D. B. M. Klaasen, \"Modeling statistical dopant fluctuations in MOS transistors\" IEEE Trans. Electron Devices, vol. ED-45, pp. 1960- 1971, 1998. [6] K. Nishinohara, N. Shigyo, and T. Wada, \"Effects of microscopic fluctuations in dopant distributions on MOSFET threshold voltage\", IEEE Trans. Electron Devices, vol. ED-39, pp. 634-639, 1992. [7] H. S. Wong and Y. Taur, \"Three-dimensional 'atomistic' simulation of discrete random dopant distribution effects in sub-0.1µm MOSFET's\", in IEDM Tech. Dig., 1993, pp. 705-708. [8] A. Asenov, \"Random dopant induced threshold voltage lowering and fluctuations in sub-0.1µm MOSFET\", IEEE Trans. Electron Devices, vol. ED-45, pp. 2505-2513, 1998. [9] A. Asenov and S. Saini, \"Suppression of random dopant-induced threshold voltage fluctuations in sub-0.1-µm MOSFET's with epitaxial and δ-doped channels\", IEEE Trans. Electron Devices, vol. ED-46, pp. 1718-1724, 1999. [10] W. J. Gross, D. Vasileska, and D. K. Ferry, \"A novel approach for introducing the electron-electron and electron-impurity interactions in particle-based simulations\", IEEE Electron Device Lett., vol. 20, pp. 463-465, 1999. [11] C. J. Wordelman and U. Ravaioli, \"Integration of a particle-particle-particle-mesh algorithm with the ensemble Monte Carlo method for the simulation of ultra-small semiconductor devices\" IEEE Trans. Electron Devices, vol. ED-47, pp. 410-416, 2000. 14 [12] N. Sano, K. Matsuzawa, M. Mukai, and N. Nakayama, \"Long-range and short range coulomb potentials in threshold characteristics under discrete dopants in sub-0.1µm Si-MOSFETs\" in IEDM Tech. Dig., 2000, pp. 275-278. [13] C.R. Arokianathan, J.H. Davies, and A. Asenov, \"Ab-initio Coulomb scattering in atomistic device simulation\", VLSI Design , vol. 8, pp. 331-335, 1998. [14] R.W. Hockney and J.W. Eastwood, \"Computer Simulation using particles\", New York: McGraw-Hill, 1981. [15] S. Barraud, P. Dollfus, S. Galdin, and P. Hesto, \"Short-range and long-range Coulomb interactions for 3D Monte Carlo device simulation with discrete impurity", + "in atomistic device simulation\", VLSI Design , vol. 8, pp. 331-335, 1998. [14] R.W. Hockney and J.W. Eastwood, \"Computer Simulation using particles\", New York: McGraw-Hill, 1981. [15] S. Barraud, P. Dollfus, S. Galdin, and P. Hesto, \"Short-range and long-range Coulomb interactions for 3D Monte Carlo device simulation with discrete impurity distribution\", Solid-State Electron., vol. 46, pp. 1061-1067, 2002. [16] W. J. Gross, D. Vasileska, and D. K. Ferry, \"Three-dimensional simulation of ultrasmall metal-oxide-semiconductor field-effect transistors: the role of the discrete impurities on the device terminal characteristics\", J. Appl. Phys., vol. 91, pp. 3737-3740, 2002. [17] N. Takimoto, \"On the screening of impurity potential by conduction electrons\", J. Phys. Soc. Jpn., vol. 14, pp. 1142-1158, 1959. [18] M. V. Fischetti and S. E. Laux, \"Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects\", Phys. Rev. B, vol. 38, pp. 9721-9745, 1988. [19] G. Masetti, M. Severi, and S. Solmi, \"Modeling of carrier mobility against carrier concentration in arsenic-, phosphorus-, and boron-doped silicon\", IEEE Trans. Electron Devices, vol. ED-30, pp. 764-769, 1983. [20] L. E. Kay and T. W. Tang, \"An improved ionized-impurity scattering model for Monte Carlo simulations\", J. Appl. Phys., vol. 70, pp. 1475-1482, 1991. [21] L. Rajaonarison, P. Hesto, J.-F. Pône, and P. Dollfus, \"Monte-Carlo simulation of submicron devices and processes\", in Proc. SISDEP'91 (Hartung-Gorre Verlag, Zürich), 1991, pp. 513-520. [22] P. Dollfus, \"Si/Si1-xGex heterostructures: electron transport and field-effect transistor operation using Monte Carlo simulation\", J. Appl. Phys., vol. 82, pp. 3911-3916, 1997. [23] T. Yamada, J. R. Zhou, H. Miyata, and D.K. Ferry, \"In-plane transport properties of Si/Si1-xGex structure and its FET performance by computer simulation\", IEEE Trans. Electron Devices, vol. ED-41, pp. 1513-1522, 1994. [24] E. Sangiorgi and M. R. Pinto, \"A semi-empirical model of surface scattering for Monte Carlo simulation of silicon n-MOSFET, IEEE Trans. Electron Devices, vol. ED-39, pp. 15 356-361, 1992 ; C. Fiegna and E. Sangiorgi, \"Modeling of high-energy electrons in MOS devices at the microscopic level\", IEEE Trans. Electron Devices, vol. ED-40, pp. 619-627, 1993. [25] S. Keith, F. M. Bufler, and B. Meinerzhagen, \"Full-band Monte Carlo device simulation of an 0.1µm n-channel MOSFET in strained silicon material\", in Proc. ESSDERC'97 (Editions Frontières, Paris), 1997, pp. 200-203. [26] P. Houlet, Y. Awano, and N. Yokoyama, \"Influence of non-linear effects on very short pMOSFET device performance\", Physica B, vol. 272, pp. 572-574, 1999. [27] D. Munteanu, G. Le Carval, and G. Guégan, \"Impact of technological parameters on non-stationary transport in realistic 50 nm MOSFET technology\", Solid-State Electron., vol. 46, pp. 10451-1050, 2002. [28] A. Asenov, R. Balasubramaniam, and A.R. Brown, \"RTS amplitudes in decananometer MOSFETs: 3-D simulation study\", IEEE Trans. Electron Devices, vol. ED-50, pp. 839- 845, 2003. 16 17 FIGURE CAPTIONS Figure 1. Electron mobility as a function of discrete dopant concentration in N-type and P-type Si resistors. Solid and dashed lines are fitting curves of experimental results given in [16] (N-type) and [17] (P-type). Symbols are our simulation results. Figure 2. Schematic view of simulated 3D MOSFETs with discrete impurities in the active region. Figure 3. Cartography", + "of discrete dopant concentration in N-type and P-type Si resistors. Solid and dashed lines are fitting curves of experimental results given in [16] (N-type) and [17] (P-type). Symbols are our simulation results. Figure 2. Schematic view of simulated 3D MOSFETs with discrete impurities in the active region. Figure 3. Cartography of current density Jx in the first sheet of cells under the gate oxide (top view). X = 0 and X = 50 nm correspond to the source/channel and channel/drain N-P junctions, respectively. The position of the three impurities present in the inversion layer is indicated by white circles (reference device). The solid and dashed lines indicate the lines along which the curves of Fig. 4 are plotted. Figure 4. Profiles of electric field Ex (a), electron concentration (b) and electron velocity vx (c) between source and drain along two lines: along Y = 38.5 nm (solid line), i.e. without impurity; and along Y = 8.5 nm (dashed line) with an impurity in XI = 23.5 nm (see Fig. 3). Figure 5. Cartography of current density Jx in the device where the impurity initially in XI = 9.5 nm (reference, Fig. 3) has been moved to XI = 43.5 nm. Figure 6. Profiles of electric field Ex (a) and electron velocity vx (b) along three lines: along Y = 38.5 nm without any impurity (solid line) and along Y = 26.5 nm with one impurity either in XI = 9.5 nm (dashed line) or in XI = 43.5 nm (dotted line). Figure 7. (a) Drain current versus gate voltage for two devices differing from one another in the X position of one impurity (VDS = 0.6 V) ; (b) Drain current versus drain voltage for the same devices (VGS = 0.6 V). Figure 8. Drive current Ion as a function of the position of one impurity in the channel 18 between source and drain along the line Y = 26.5 nm (Nimp = 37, circles). The graph also shows the results obtained by removing this impurity (Nimp = 36, closed diamond) and by adding (Nimp = 38, open square) or removing (Nimp = 36, open diamond) another impurity at XI = 23.5 nm. Figure 9. Cartography of current density Jx in two devices where one impurity has been removed (a) or added (b) at XI = 23.5 nm (to be compared with the reference, Fig. 3). Figure 10. Drain current versus gate voltage for three devices differing from one another in the number of impurities in the version layer: the three curves correspond to the devices whose current cartography is shown in Fig. 3 (solid line), Fig. 9a (dotted line) and Fig. 9b (dashed line) (VDS = 0.6 V). Figure 11. Cartography of current density Jx in three devices differing from one another in the vertical position of surface impurities (with Nimp = 37). When compared to the reference (Fig. 3), three surface impurities are buried 3.5 nm below the surface (a) or four buried impurities are moved to the surface (b and c) at different positions. Figure 12. Drain current versus", + "another in the vertical position of surface impurities (with Nimp = 37). When compared to the reference (Fig. 3), three surface impurities are buried 3.5 nm below the surface (a) or four buried impurities are moved to the surface (b and c) at different positions. Figure 12. Drain current versus gate voltage (at VDS = 0.6 V) for four devices differing from one another in the vertical position of some impurities, i.e. in the number of surface impurities which is either 3 (reference, solid line), 0 (dotted line) or 7 (dashed lines). These devices correspond to the current cartographies of Figs. 3 (circles), 11a (squares), 11b (triangles) and 11c (diamonds). Figure 13. Bar chart of Ion values of all simulated devices. The dashed line indicates the average value on I . 19 Dollfus et al. Figure 1 100 1000 1014 1015 1016 1017 1018 1019 N type / simulation P type / simulation N type / experiment P type / experiment Electron Mobility (cm2/Vs) Doping Concentration (cm-3) 20 Dollfus et al. Figure 2 W L ZJ tOX X Y Z 21 Dollfus et al. Figure 3 10 20 30 40 50 0 10 20 30 40 50 0 Length X (nm) Width Y (nm) 0 1 2 3 4 ( kA/cm2 ) 22 Dollfus et al. Figure 4 -500 -400 -300 -200 -100 0 100 200 -10 0 10 20 30 40 50 60 Electric Field, E X (kV/cm) Distance along the channel, X (nm) XI = 23.5 nm (a) 1018 1019 1020 -10 0 10 20 30 40 50 60 Electron Concentration (cm -3) Distance along the channel, X (nm) XI = 23.5 nm (b) 0 0.5 1 1.5 2 2.5 3 -10 0 10 20 30 40 50 60 Electron Velocity, v X (107 cm/s) Distance along the channel, X (nm) XI = 23.5 nm (c) 23 Dollfus et al. Figure 5 10 20 30 40 50 0 10 20 30 40 50 Length X (nm) Width Y (nm) 0 24 Dollfus et al. Figure 6 -500 -400 -300 -200 -100 0 100 200 -10 0 10 20 30 40 50 60 Electric Field, E X (kV/cm) Distance along the channel, X (nm) XI = 43.5 nm XI = 9.5 nm (a) 0 0.5 1 1.5 2 2.5 3 -10 0 10 20 30 40 50 60 Electron Velocity, v X (107 cm/s) Distance along the channel, X (nm) XI = 43.5 nm XI = 9.5 nm (b) 25 Dollfus et al. Figure 7 0 5 10 15 20 25 30 35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Drain Current, I D (µA) Gate Voltage, V GS (V) impurity near drain-end (XI = 43.5 nm) impurity near source-end (XI = 9.5 nm) (a) 0 5 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 XI = 9.5 nm XI = 43.5 nm Drain Current, I D (µA) Drain Voltage, V DS (V) VGS = 0.6 V (b) 26 Dollfus et al. Figure 8 29.5 30 30.5 31 31.5 32 0 10 20 30 40", + "30 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 XI = 9.5 nm XI = 43.5 nm Drain Current, I D (µA) Drain Voltage, V DS (V) VGS = 0.6 V (b) 26 Dollfus et al. Figure 8 29.5 30 30.5 31 31.5 32 0 10 20 30 40 50 Drive Current, I on (µA) Impurity Position, X I (nm) Nimp = 37 Nimp = 38 Nimp = 36 reference device 27 Dollfus et al. Figure 9 10 20 30 40 50 0 10 20 30 40 50 Length X (nm) Width Y (nm) 0 10 20 30 40 50 0 10 20 30 40 50 0 Length X (nm) Width Y (nm) 28 Dollfus et al. Figure 10 0 5 10 15 20 25 30 35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Drain Current, I D (µA) Gate Voltage, V GS (V) Nimp = 37 Nimp = 38 Nimp = 36 29 Dollfus et al. Figure 11 10 20 30 40 50 0 10 20 30 40 50 Length X (nm) Width Y (nm) 0 10 20 30 40 50 0 10 20 30 40 50 0 Length X (nm) Width Y (nm) 10 20 30 40 50 0 10 20 30 40 50 0 Length X (nm) Width Y (nm) 30 Dollfus et al. Figure 12 0 5 10 15 20 25 30 35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Drain Current, I D (µA) Gate Voltage, V GS (V) no surface impurity reference 7 surface impurities Nimp = 37 31 Dollfus et al. Figure 13 24 26 28 30 32 34 Drive Current, I on (µA) Device ", + "A flexible 300 mm integrated Si MOS platform for electron- and hole-spin qubits exploration R. Li1, N. I. Dumoulin Stuyck1,2, S. Kubicek1, J. Jussot1, B. T. Chan1, F. A. Mohiyaddin1, A. Elsayed1,3, M. Shehata1,3, G. Simion1, C. Godfrin1, Y. Canvel1, Ts. Ivanov1, L. Goux1, B. Govoreanu1 & I. P. Radu1 1Imec, B-3001 Leuven, Belgium, email: roy.li@imec.be 2Department of Materials Engineering (MTM), KU Leuven, B-3001 Leuven, Belgium 3Department of Physics and Astronomy, KU Leuven, B-3001 Leuven, Belgium Abstract—We report on a flexible 300 mm process that optimally combines optical and electron beam lithography to fabricate silicon spin qubits. It enables on-the-fly layout design modifications while allowing devices with either n- or p-type ohmic implants, a pitch smaller than 100 nm, and uniform critical dimensions down to 30 nm with a standard deviation ~ 1.6 nm. Various n- and p-type qubits are characterized in a dilution refrigerator at temperatures ~ 10 mK. Electrical measurements demonstrate well-defined quantum dots, tunable tunnel couplings, and coherent spin control, which are essential requirements for the implementation of a large-scale quantum processor. I. INTRODUCTION Silicon quantum dot qubit systems are strong candidates for large-scale quantum processors. Long coherence times and high- fidelity operations have been demonstrated in fundamental qubit gates [1]. Although these devices are fabricated in lab environments, their structures are compatible with state-of-the-art silicon technologies. Upscaling a quantum processor crucially relies on the performance of each individual physical qubit [2]. However, a detailed understanding of the device and material properties at qubit operation temperature is still lacking. Efforts in scaling up silicon quantum processors beyond the few-qubit scale are limited by unpredicted parasitic effects. In this article, we discuss our design cycles towards a better understanding of cryogenic material properties and present our latest developments on qubit fabrication and characterization. Fig. 1 depicts the concept of the design cycle, which is built upon the close link between fabrication, characterization, and device modelling. We develop a 300 mm process flow for qubit specific integration that produces high volume and uniform devices. Furthermore, it allows rapid design updates and enables the incorporation of complex structures, different gate materials, and either n- or p-type implants as ohmics. At 300 K, we perform standard semiconductor characterizations on both dedicated metrology structures and qubit structures for process control and for studying the influence of fabrication parameters. Cryogenic characterization consists of qubit measurements in a dilution refrigerator at temperatures in the milli-Kelvin range. We show controlled charge and spin operations, which can be correlated with process, design, and room temperature data. Qubits are simulated with a multi-physics model, which can be calibrated with cryogenic hardware data [3]. Finally, the updated model guides the design for the next cycle. II. DEVICE FABRICATION The fabrication is based on a 300 mm process (Fig. 2(a)). Relaxed size features, including zero markers, ohmic junctions, spin-control stripline antennas or micro-magnets, and fanout metal are defined by optical lithography. The pitch-critical qubit gate structures are patterned by electron beam lithography (EBL) on 300 mm wafers in predefined qubit areas (Fig. 2(b)) [4]. We choose EBL here as a short", + "size features, including zero markers, ohmic junctions, spin-control stripline antennas or micro-magnets, and fanout metal are defined by optical lithography. The pitch-critical qubit gate structures are patterned by electron beam lithography (EBL) on 300 mm wafers in predefined qubit areas (Fig. 2(b)) [4]. We choose EBL here as a short turnaround replacement for advanced optical lithography, which would limit rapid device redesign and process. The EBL modules are fully co-integrated with optical lithography modules, in a single integrated process, which includes all needed blocks for qubit functionality. A specific back end of line (BEOL) passivation module and via contacting are developed, compatible with sample preparation requirements for cryogenic characterizations (Fig. 2(c-d)). In the qubit region, we have the flexibility to incorporate different structures across a 300 mm wafer, from a double quantum dot device to a one-dimensional (1D) array (Fig. 3). We adopt a 3-layer overlapping gate structure with a pitch of 100 nm or below, allowing tight confinement potential around the quantum dots. Furthermore, the thickness of each gate metal is 20 ~ 30 nm and the isolation dielectric is ~ 5 nm. This reduces the topography influence so that each gate layer has similar coupling to the quantum dots underneath (Fig. 4). The geometry of the gates greatly affects the qubit operations and interactions. The width and pitch of the EBL gates is monitored by critical dimension scanning electron microscope (CDSEM) across the wafer. As an example, we monitor the confinement gate gap that defines the lateral size of all quantum dot in a 1D array (Fig. 5). We record the dimensions of the resist after EBL and gate after etching because they are important design parameters. Their 3σ are ~ 5 nm, confirming high EBL pattern uniformity. To summarize, the established fab recipes and inline monitoring guarantee uniformity and reproducibility, while the EBL qubit gates allow fast design turnaround. The whole integration makes systematic design variations possible, which is crucial for the following study of material properties and design criteria. III. 300 K CHARACTERIZATION The 300 mm fab wafers are characterized with automated probe station systems. We incorporate different metrology structures across the wafer to extract material properties and monitor process steps. Fig. 6(a) shows the IdVg studies of metrology transistors with different channel lengths on different gate layers. Long channel transistors show a systematic upward shift in the threshold voltage with higher gate layers, which have thicker dielectrics. However, this trend is not clear for short channel transistors due to short channel effects. Nonetheless, the narrow cumulative distribution function (CDF) of the threshold voltage signifies high process uniformity (Fig. 6(b)). Qubits are tested at room temperature with customized probe cards (Fig. 7(a)). For EBL gate patterning, we can reliably achieve 30 nm spacing without shorts (Fig. 7(d)). Smaller spacings can be achieved with EBL recipes for smaller pitches. We further study IdVg on single quantum dot structures (Fig. 8). The right barrier (RB) has a slightly larger CDF span than that of the left barrier (LB) with the same design size. It hints at EBL proximity or etching", + "spacings can be achieved with EBL recipes for smaller pitches. We further study IdVg on single quantum dot structures (Fig. 8). The right barrier (RB) has a slightly larger CDF span than that of the left barrier (LB) with the same design size. It hints at EBL proximity or etching loading effects as RB is closer to the fanout regions of the qubit gates. 300 K characterization allows us to extract material properties and design topography related peculiarities systematically. This can be further studied for correlations to cryogenic results [5]. IV. CRYOGENIC CHARACTERIZATION To verify qubit device designs and study qubit control related properties, we characterize various qubit structures in a dilution refrigerator with a base temperature lower than 10 mK. We study single charge control with a similar gate structure as Fig. 4(b). We use barrier gates LB and RB to isolate a quantum dot along the channel induced by the top gate ST (Fig. 9(a)). This structure is also called single electron transistor (SET) for nMOS (or single hole transistor (SHT) for pMOS). In the small bias condition (source-drain bias Vsd < 1 mV), only a single electron (hole) can pass through the dot each time because of Coulomb repulsion. This structure can be described by a classical constant interaction model, and the equivalent circuit is shown in Fig. 9(b) [6]. We measure the dot conductance while sweeping LB and RB. The ~ 45o high conductance lines in Fig. 9(c) and (d) confirm a quantum dot formed between LB and RB. Fig. 10(a) shows the Coulomb current oscillations in a SET, denoting consecutive addition of individual electrons to the dot. On sweeping Vsd, the Coulomb oscillations develop into Coulomb diamonds, and the regular diamond patterns signify a well-defined quantum dot (Fig. 10(b)) [6]. Like a single dot, a double dot can be induced by stacking more gates. We define a double dot system with a similar structure as Fig. 3(b), where the quantum dots are defined under plunger gates 1 and 2 (P1 and P2). The schematic cross-section and the circuit model are shown in Fig. 11(a) and (b), respectively. For both nMOS and pMOS double dot devices, we can demonstrate separate control of the double dot potential with P1 and P2, and highly tunable inter-dot tunnel coupling with the barrier gate B2 (Fig. 11 (c)) [5,6]. The previous measurements characterize quantum dots in the many charge regime. However, single spin qubits generally require single charges in the quantum dots. In Fig. 12, we show that the quantum dots can be emptied to the very last charge. We measure a pMOS double dot device similar to Fig. 4, and the schematic cross-section is shown in Fig. 12(a). We operate the SHT dot as a potentiometer to sense the charges in dot 1 and 2 (equivalent circuit shown in Fig. 12(b)) [6]. By sweeping P1 and P2 voltages, we can empty the holes in dot 1 and 2 to the very last (Fig. 12(c)). We correct the gate crosstalk by remapping P1 and P2 onto a virtual gate space, allowing separate", + "dot 1 and 2 (equivalent circuit shown in Fig. 12(b)) [6]. By sweeping P1 and P2 voltages, we can empty the holes in dot 1 and 2 to the very last (Fig. 12(c)). We correct the gate crosstalk by remapping P1 and P2 onto a virtual gate space, allowing separate control of the potential on dot 1 and 2 [7]. This is shown by the almost vertical and horizontal transition lines in Fig. 12(d). Furthermore, we demonstrate tunable inter-dot tunnel coupling in the single hole state (Fig. 13). This is required for robust two qubit gates [8]. We now access the spin qubit operations. We start with pMOS qubits, as the strong spin-orbit coupling (SOC) of holes allows electrical spin control without the need for extra structures such as stripline antennas or micromagnets [9]. We induce a double quantum dot system and tune it to weak tunnel coupling regime, similar to Fig. 11(c) bottom left. With increased |Vsd|, the conductance point expands to a bias triangle (Fig. 14). In some charge configurations, the bottom of the bias triangle could be missing, as shown in Fig. 14(a) right. This is due to Pauli spin blockade (PSB) as spins in the double dots form a triplet state that blocks the current flow (Fig. 14(b) right) [6]. In the PSB regime, we can readout the spin state, as higher current means one of the triplet spins has been flipped. With SOC, we control the spin by applying microwave (MW) burst to the quantum dot gate. When the MW frequency equals the spin Zeeman splitting by the magnet field, spins can be flipped, as shown in Fig. 15 [6]. Finally, we demonstrate coherent spin control with Rabi oscillations [9]. By sweeping the MW burst time, the spin oscillates up and down with a Rabi frequency ~ 18 MHz, as shown in Fig. 16(a). We further show the quadratic dependency of Rabi frequency with MW power, as the spin control speed is proportional to the MW amplitude (Fig. 16(b)). V. SUMMARY & FUTURE WORK Scaling up the qubit number and further improving the qubit performance are the main research focuses for silicon quantum dot qubit systems. We address them by developing a 300 mm process for qubit device integrations. Through this process, we incorporate different device designs, achieve highly uniform gates with 3σ ~ 5 nm, and demonstrate almost 100% device yield for 30 nm gate spacings. We further verify the process flow and device designs by cryogenic measurements, demonstrating well- defined single charges, highly tunable tunnel couplings, and coherent spin operations. Future efforts involve incorporating more materials and structures into the qubit devices in the fab process and improving cryogenic characterization throughputs by low temperature multiplexing and clever device designs. Through this learning cycle, knowledge on relevant material properties can be effectively collected, which will pave the way for large-scale silicon spin quantum processors. REFERENCES [1] L. M. K Vandersypen et al., npj Quantum Information 3, 34, 2017. [2] N. Moll et al., Quantum Sci. Technol. 3, 030503, 2018. [3] F. A. Mohiyaddin et al., Proc.", + "relevant material properties can be effectively collected, which will pave the way for large-scale silicon spin quantum processors. REFERENCES [1] L. M. K Vandersypen et al., npj Quantum Information 3, 34, 2017. [2] N. Moll et al., Quantum Sci. Technol. 3, 030503, 2018. [3] F. A. Mohiyaddin et al., Proc. of IEDM, pp. 39-5, 2019. [4] B. Govoreanu et al., Proc. of Silicon Nanoelectronics Workshop, 2019. [5] R. Pillarisetty et al., Proc. of IEDM, pp. 31-5, 2019. [6] R. Hanson et al., Rev. Mod. Phys. 79, 1217, 2007. [7] C. Volk et al., npj Quantum Information 5, 29, 2019. [8] M. D. Reed et al., Phys. Rev. Lett. 116, 110402, 2016 [9] N. W. Hendrickx et al., Nature. 577, 487, 2020. Fig. 1. Flowchart representation of the design cycle. The cycle is designed towards a detailed understanding of material properties at cryogenic temperatures for qubit up-scaling. It relies on close links between integration, characterization, and modelling. The key requirements for each part are summarized in the sub-tables. Fig. 2. (a) Schematic of the hybrid 300 mm fab integration flow for qubit devices. Qubit gates are patterned by EBL and surrounding relaxed-size structures are patterned by optical lithography. (b) SEM image of the qubit gate region. (c) Schematic of the BEOL module and contacting vias stopping on an EBL gate and ohmic. (d) Cross-section SEM image of the via contact between the optical lithography metal and the EBL qubit gate. Fig. 3. SEM images of EBL qubit gates designed for (a) double quantum dots under P1 and P2 with a charge sensor dot between LB and RB; (b) triple quantum dots under P1, P2, and P3; and (c) a 1D quantum dot array. Fig. 4. (a) Schematic of a qubit device similar to Fig. 3(a). Gate layers 1 / 2 / 3 are shown as green / red / blue. Dark blue denotes charge carriers. (b) and (c) are the TEM cross-section at the charge sensor dot region and the double dot region, respectively. Fig. 5. CDSEM measurement on the confinement gate gap. (a) and (c) show resist after EBL and the gate after etching, respectively. (b) and (d) show the histogram of over 50 CDSEM across the 300 mm wafer on the structures in (a) and (c), respectively. Fig. 6. (a) IdVg curves for metrology nMOS transistors with channel length L = 1 and 10 µm on different gate layers. (b) CDF of the threshold voltages. For long channel transistors, threshold increases with gate layers, while this trend is less clear for short channel transistors. Fig. 7. (a) A customized probe card aligned to the bond pads of a qubit device. (b) The gate structure undergoes the probe card testing. Gate spacings are varied across the wafer, and inter-gate shorts are measured. (c) Probe card result matrix. Each box denotes a device. Spacings are changed column-wise. From left, the spacings are 20~25 nm, 25~30 nm, 30~35 nm, and repeat. Green (yellow) means no short (short). (d) The yield rate table. Fig. 8. (a) IdVg curves for the SET gates. Inserts show the", + "(c) Probe card result matrix. Each box denotes a device. Spacings are changed column-wise. From left, the spacings are 20~25 nm, 25~30 nm, 30~35 nm, and repeat. Green (yellow) means no short (short). (d) The yield rate table. Fig. 8. (a) IdVg curves for the SET gates. Inserts show the structures and sweeping gates are false colored red. The non- sweeping gates are kept at 3.5 V. (b) CDF of the threshold voltages, which are defined by linear fitting to the sub-threshold regions to increase the sensitivity to gate topographies. Fig. 9. (a) Schematic cross- section of a single quantum dot defined by similar gates as Fig. 4(b). (b) Equivalent circuit of a single dot. (c) and (d) show the conductance through the single dot by sweeping LB and RB for a nMOS and pMOS dot, respectively. The ~ 45o high conductance lines indicate the dot is equally coupled to LB and RB. Fig. 10. (a) Coulomb oscillations of a SET measured at Vsd = 1 mV. (b) Coulomb diamonds of the SET. The differential conductance is measured by standard lock-in technique with an ac excitation dVsd = 100 µV. Fig. 11. (a) Schematic cross-section of a double dot defined by similar gates as Fig. 3(b). (b) Equivalent circuit of a double dot. (c) Tunable tunnel coupling between the double dot for nMOS (upper) and pMOS (lower) devices. The plunger gates P1 and P2 control the potential of dot 1 and 2, respectively. B2 controls the inter-dot coupling. In the weak coupling regime, dot 1 and 2 are separated. Conductance maps show isolated points when energy levels between dots are aligned. In the medium coupling regime, levels of double dots begin to hybridize. Conductance maps show homeycomb patterns. In the strong coupling regime, two dots merge into one. Conductance maps show single dot features similar to Fig. 9 (c-d). Fig. 12. (a) Schematic cross- section of a double dot with a charge sensor. The gate structure is similar to Fig. 4(c). (b) Equivalent circuit. (c) Charge sensing map on a pMOS device. Brackets denote the hole numbers in dot 1 and 2. (d) Remap of (c) in the virtual gate space. Crosstalk between P1 and P2 is removed. Fig. 13. (a) Simulated energy diagram of inter-dot transitions with different coupling strength. (b-d) are zoom in maps in Fig. 12(d) at the (1,0) - (0,1) transition region. By tuning the B1 bias, the inter-dot tunnel coupling can be changed from weak to strong with B1, resembling the patterns in (a). Fig. 14. (a) Bias triangles of a pMOS double dot system in weak coupling regime like that of Fig. 11 but with higher |Vsd|. For Vsd = -2 mV, the triangle bottom is missing due to spin Pauli blockade: When there is an unpaired spin in dot 1, the upcoming spin from dot 2 will be blocked if they form a triplet state, as shown in (b) right. For Vsd = 2 mV, the extra spin enters dot 1 from the source reservoir, and hence a singlet state can always be loaded, as", + "unpaired spin in dot 1, the upcoming spin from dot 2 will be blocked if they form a triplet state, as shown in (b) right. For Vsd = 2 mV, the extra spin enters dot 1 from the source reservoir, and hence a singlet state can always be loaded, as shown in (b) left. Fig. 15. Double dot current in the PSB region as a function of external magnetic field and the frequency of MW burst applied on the double dot gate. With spin-orbit interaction, the spin of holes can be flipped when the MW photon frequency equals the spin Zeeman energy. Spin flipping lifts the PSB, increasing the current through the double dot. The high current line remarks the linear relation between magnetic field and MW frequency for on- resonance spin flip. Fig. 16. (a) Coherent spin control with Rabi oscillations. When the MW frequency is tuned to be on-resonance with the spin Zeeman energy, the spin direction can be controlled coherently by changing the MW burst time. The oscillation of spin direction is reflected by the oscillation of current in the PSB regime. (b) Power dependency of Rabi oscillations. The Rabi frequency depends on the MW amplitude. Therefore, the peaks or dips of the Rabi oscillations shift quadratically when stepping the MW power.", + "A sparse spin qubit array with integrated control electronics J. M. Boter1,2,*, J. P. Dehollain1,2,3,*, J. P. G. van Dijk1,2, T. Hensgens1,2, R. Versluis1,4, J. S. Clarke5, M. Veldhorst1,2, F. Sebastiano1, and L. M. K. Vandersypen1,2,5,† 1QuTech, Delft University of Technology, The Netherlands, 2 Kavli Institute of Nanoscience, Delft University of Technology, The Netherlands, 3School of Mathematical and Physical Sciences, University of Technology Sydney, Australia 4Netherlands Organisation for Applied Scientific Research (TNO), The Netherlands, 5Components Research, Intel Corporation, USA *These authors contributed equally to this work †Mail: Lorentzweg 1, 2628 CJ Delft, The Netherlands, l.m.k.vandersypen@tudelft.nl Abstract—Current implementations of quantum computers suffer from large numbers of control lines per qubit, becoming unmanageable with system scale up. Here, we discuss a sparse spin-qubit architecture featuring integrated control electronics significantly reducing the off-chip wire count. This quantum- classical hardware integration closes the feasibility gap towards a CMOS quantum computer. I. INTRODUCTION Semiconductor spin qubits [1, 2] are an attractive platform for large-scale quantum computers, due to their potential compatibility with well-established semiconductor manufacturing processes. In the last decade we have witnessed tremendous progress in the development of spin-qubit hardware [3-8] and significant interest and contribution of the semiconductor industry into spin-qubit research [9-11]. Therefore, the open questions surrounding the challenge of scaling-up [12] have become timely and highly relevant. One of the main issues in common with all nanoelectronic qubits is that current implementations require at least one external control line for every qubit. The small pitch of quantum dots (Fig. 1) permits extremely dense qubit arrays but aggravates the interconnect challenges. Existing proposals for dense 2D spin qubit arrays [13, 14] assume either a device density or material homogeneity that remains to be achieved. Another approach involves a network architecture, where qubits are arranged in small-cluster nodes, interconnected by long-range entanglement distribution channels, with the goal of creating space for easing the density requirements of the interconnects [12]. The feasibility of implementing quantum error correction protocols using this approach has been thoroughly analyzed [15], but the description of the physical implementation is largely missing. Here, we present a design of a sparse two- dimensional array whereby classical electronics integrated locally with the quantum hardware is used to minimize the need for off-chip interconnects and hence with a scalable Rent’s exponent [16]. We first describe the components of the array and the implementation of quantum gates and measurements, followed by a description of the control electronics required to operate the qubits in the array and correct errors via the surface code approach [17]. We then analyze how this implementation of locally integrated electronics reduces the number of connections at the quantum plane boundary, and the required footprint of such components. Finally we provide a discussion of some of the technological considerations, potential challenges and options for solving them. II. ARRAY DESIGN We propose a quantum computing architecture consisting of a two-dimensional array of electron-spin qubits using linear arrays of gate electrodes (Fig. 1) arranged to form a square lattice of electrostatically defined quantum dots with nearest- neighbor connectivity. In conventional spin qubit designs every quantum", + "for solving them. II. ARRAY DESIGN We propose a quantum computing architecture consisting of a two-dimensional array of electron-spin qubits using linear arrays of gate electrodes (Fig. 1) arranged to form a square lattice of electrostatically defined quantum dots with nearest- neighbor connectivity. In conventional spin qubit designs every quantum dot, with a typical pitch smaller than 100 nm, hosts a qubit. The proposed design uses a sparse qubit array with the qubits separated by ~12 μm and the vertices connected via electron shuttling channels to transport electrons to and from interaction regions. The array’s sparseness enables the integration of sample-and-hold circuits alongside the quantum dot circuitry allowing to offset the inhomogeneities in the potential landscape across the array by independent DC biasing while sharing the majority of control signals for qubit operations across the array. The latter allows for a significant reduction in the number of connections at the quantum plane boundary. As detailed in Fig. 2, we start from a 22 mm × 33 mm (726 mm2) die. The qubits are defined in the quantum plane, a section of the die consisting of 𝑀× 𝑀 modules, each containing 𝑁× 𝑁 unit cells. The unit cell is the smallest operational unit, containing four qubits along with all the elements required to operate them, as described in Fig. 3. Qubits remain at the vertices of the lattice while idle and are shuttled to the operation regions between the vertices in order to perform single- and two-qubit operations as well as readout and initialization. III. DC BIASING Fig. 4 shows circuit schematics of locally integrated sample- and-hold circuits providing individual DC biasing of all the control gates, which total 64 gates per unit cell. Gate voltages within a unit cell are updated sequentially via four local demultiplexers that each distribute DC voltages generated remotely (i.e., outside the quantum plane) to 16 local capacitors connected to the gates. All demultiplexers within a module share the same input DC biasing signal, and all demultiplexers in the quantum plane share the same address bus (see Fig. 4(f)). The demultiplexers are enabled sequentially and in turn sequentially update each gate. In this way, all modules are updated in parallel and therefore one module refresh cycle is required to refresh the entire qubit array. We define two bias voltage resolutions, based on the gate functionalities. Gates acting as barriers to shuttling channels only require a resolution sufficient to maintain an electron in a quantum dot and therefore we can afford a coarse resolution of 1 mV. All other plunger and barrier gates require a resolution of 1 μV [12]. The minimum hold capacitance required to achieve the coarse resolution is ~0.16 fF (limited by the electron charge 𝑒/∆𝑉), while the fine resolution requires ~14 pF (limited by thermal noise 𝑘𝐵𝑇/𝐶, assuming power dissipation from the local electronics raises the operating temperature to 1 K). The gate voltage refresh rate will be set by the current leakage of the hold circuit and the time required to update each gate, which in turn will set the module size (i.e., the", + "noise 𝑘𝐵𝑇/𝐶, assuming power dissipation from the local electronics raises the operating temperature to 1 K). The gate voltage refresh rate will be set by the current leakage of the hold circuit and the time required to update each gate, which in turn will set the module size (i.e., the number of unit cells, and therefore total gates, which can be sequentially updated). IV. SIGNALS FOR QUBIT OPERATIONS All the qubit operations are performed by shuttling the qubits to the operation regions and applying pulsed signals to the appropriate gates to perform the operations. The shuttle channels are defined by a linear array of gates (blue gates in Fig. 3), along which a traveling wave potential can trap and shuttle an electron. The traveling wave potential is generated using four phase-shifted sinusoidal signals on four consecutive gates (different shades of blue in Fig. 3), with the signals being reused every fifth gate. The shuttling signals are always on and the phase shifts control the direction of shuttling. With the use of a barrier gate (Fig. 3(b)), an electron can be forced to tunnel into a shuttle channel. The traveling wave potential is made large enough to overcome the inhomogeneities in the potential landscape, eliminating the need to apply DC biasing on the shuttling gates. Single-qubit gates are performed by applying a microwave pulse to the control gate labelled MW in Fig. 3(c). A pair of micromagnets in this operation region provides magnetic field gradients required to perform electric dipole spin resonance (EDSR) [18]. A two-qubit gate is performed by pulsing the control gate labelled J, to activate an exchange interaction between electrons underneath the adjacent gates. In order to apply the AC signals on gates that also require DC biasing, we make use of a complementary switching circuit (see 𝜑𝐴𝐶 and 𝜑𝐴𝐶 ̅̅̅̅̅ in Fig. 4(b)). The surface code is sustained using a cyclic sequence of pulsed signals within a unit cell, with the same sequence performed in parallel across all unit cells in the entire array. A set of remote pulsed voltage sources is used to generate the required cyclic pulsed signals at each gate (i.e., one source per pulsed gate in a unit cell). Logic gates in the surface code with lattice surgery are achieved by creating defects in the lattice. We implement these defects by preventing shuttling of a subset of data qubits via locally integrated switches. V. READOUT Qubit readout is performed at the operation region shown in Fig. 3(c). A readout quantum dot connected to source/drain ohmic contacts is used for charge sensing and spin readout is achieved via spin-to-charge conversion based on Pauli spin blockade [2]. Additionally, the ohmics in this region provide electrons that are shuttled to the unit cell vertices to initialize the array. The drain contacts of all readout dots in a module are connected to a single line at the quantum plane boundary, and readout is performed sequentially across the unit cells of each module, while the modules are read out in parallel. This is achieved by sequentially pulsing every sensor plunger", + "The drain contacts of all readout dots in a module are connected to a single line at the quantum plane boundary, and readout is performed sequentially across the unit cells of each module, while the modules are read out in parallel. This is achieved by sequentially pulsing every sensor plunger in a module to the low-impedance, electrostatically sensitive regime, while all other sensors in the module are at high- impedance (i.e., Coulomb blockade). The sequential control of the plungers in a module is achieved using a global readout demultiplexer that can be shared between all modules across the entire array. VI. LINE SCALING The signal routing we have described (as summarized in Table 1), enabled by the described DC biasing scheme, allows for a very efficient scaling of the ratio of connections needed at the unit cell level to connection outputs at the quantum plane boundary. Considering that the total number of gates scales with the number of qubits (4M2N2), we now discuss how the operation schemes described above allow scaling down the number of connections at the quantum plane boundary. The sparse array with sample-and-hold circuits provides independent DC biasing with O(M2+N) lines at the quantum plane boundary. All pulsed and microwave control signals needed to sustain the surface code, can be shared across every unit cell in the entire array. This amounts to a constant number of 58 lines at the quantum plane boundary irrespective of the number of qubits. The signals used to control the switches that deactivate data qubits for the logical qubit implementation scheme, are arranged in a crossbar fashion across the entire quantum plane, reducing the number of lines for this purpose to as few as O(MN) at the quantum plane boundary. In practice, we propose to use x crossbars over the entire array, in order to allow for x defects to be simultaneously created and manipulated, bringing the line scaling to O(xMN). By using decoding to address the readout plungers per module, the sequential readout scheme obtains a line scaling at the quantum plane boundary as O(M2+log(N)). At the boundary of the quantum plane, Rent’s exponent can thus be as low as p=0.5. VII. FOOTPRINT We now consider the footprint requirements of the control electronics that need to be locally integrated in the quantum plane, and the wire density at various levels. The bulk of the footprint will be taken up by the capacitors required for the sample-and-hold scheme. Coarse resolution is required for 32 gates and another 32 gates require fine resolution, which comprise a total capacitance per unit cell of ~450 pF. Assuming ~1 pF/μm2 (using state-of-the-art deep- trench capacitor technology [19]), we estimate a total capacitor footprint of ~450 μm2. In addition, we modelled a demultiplexer circuit using 40-nm technology, extrapolated to 28-nm technology and obtained an estimate of the total footprint of the DC biasing and readout demultiplexers of ~60 μm2 per unit cell. This adds to a total footprint per unit cell of ~510 μm2, which allows to set the qubit pitch to 𝑑≥12 μm. Assuming a 50 nm pitch", + "28-nm technology and obtained an estimate of the total footprint of the DC biasing and readout demultiplexers of ~60 μm2 per unit cell. This adds to a total footprint per unit cell of ~510 μm2, which allows to set the qubit pitch to 𝑑≥12 μm. Assuming a 50 nm pitch between gate electrodes (Fig. 1), this would require linear arrays of 240 gate electrodes per lattice arm. A unit cell has an area 4𝑑2 ≈576 μm2 and a perimeter 8𝑑≈96 μm. O(102-103) wires pass through the unit cell perimeter, using multiple interconnect layers. The area and perimeter for a module are (2𝑑𝑁)2 and 8𝑑𝑁, respectively, and the quantum plane has an area and a perimeter of (2𝑑𝑁𝑀)2 and 8𝑑𝑁𝑀, respectively. For a total of 220 (≈106) qubits, the total area covered by the quantum plane is ~151 mm2, leaving ~575 mm2 of space remaining in the die, which can be used to implement classical control circuits and to bring the wire count going off-chip, typically the real bottleneck for Rent’s rule, to well below the wire count at the quantum plane boundary by means of additional levels of multiplexing. VIII. DISCUSSION AND OUTLOOK Very-large scale spin qubit devices will ultimately be based on a trade-off of a large number of considerations. With this proposal we explore the extreme sparse approach, with single qubits placed at the nodes of the shuttling channels. Different from some existing proposals, this approach does not make strong assumptions on the potential landscape homogeneity or the density with which transistors and qubits can be integrated, but it does assume that spins can be shuttled over 10 µm distances with very high fidelity. It should also be possible to design a similar integrated electronic scheme for architectures with larger qubit-cluster nodes, for which it has been shown that the fidelity requirements of the shuttling channels are more relaxed [15]. We also assume that magnetic field inhomogeneities and g-factor variations can be overcome by individual dc tuning. In this work we have focused on the reduction of the number of control lines at the quantum plane boundary, as well as on the footprint of the classical control electronics, a key first step to assess the feasibility of implementing sparse spin qubit architectures, which motivates future work into addressing the following open issues. Distributing all signals for qubit operations across the entire qubit array requires careful design for minimizing crosstalk, along with to estimate the total line capacitance, which will affect clock speeds and required source power. There will be a large number of switches, used to separate the cycles of DC biasing and qubit operation on the applied gate voltages and to perform lattice surgery. The power dissipated by these switches can be significant and will be a factor in considering the clock rates of the system and the achievable operating temperature. It is most likely that both the surface code cycle rate and the size of the array will be limited by the number of sequential readouts required, since this is the most time consuming of all operations. Some", + "considering the clock rates of the system and the achievable operating temperature. It is most likely that both the surface code cycle rate and the size of the array will be limited by the number of sequential readouts required, since this is the most time consuming of all operations. Some degree of parallel readout can be implemented by amplitude modulation or frequency modulation. If that is not sufficient, smaller readout modules can be defined, each consisting of a subset of unit cells that are read sequentially. This comes at the expense of an increased number of readout connections. All things considered, this proposal provides an appealing outlook for the long-term implementation of larger scale quantum computing chips, and provides guidance for near- term research at the quantum, classical and integrated levels. REFERENCES [1] D. Loss, and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). [2] F.A. Zwanenburg et al., Rev. Mod. Phys. 85, 961 (2013). [3] E. Kawakami et al., Nat. Nano 9, 666 (2014). [4] M. Veldhorst et al., Nat. Nano 9, 981 (2014). [5] J. Yoneda et al., Nat. Nano 13, 102 (2018). [6] T. F. Watson et al., Nature 555, 633 (2018). [7] X. Xue et al., Phys. Rev. X 9, 021011 (2019). [8] W. Huang et al., Nature 569, 532 (2019). [9] R. Maurand et al., Nat. Comm. 7, 13575 (2016). [10] P. Galy et al., IEEE J. of the Electron Devices Society 6, 594 (2018). [11] R. Pillarisetty et al., in 2018 IEEE Int. Elec. Dev. Meet. (IEDM) (IEEE, New York, 2018), pp. 6.3.1-6.3.4 [12] L. M. K. Vandersypen et al., npj Quantum Inf. 3, 34 (2017). [13] M. Veldhorst et al., Nat. Comm. 8, 1766 (2017). [14] R. Li et al., Sci. Adv. 4, eaar3960 (2018). [15] B. Buonacorsi et al., Quant. Sci. Tech. 4, 025003 (2019). [16] D. P. Franke et al., Microprocessors and Microsystems 67, 1 (2019). [17] C. Horsman et al., New J. of Phys. 14, 123011 (2012). [18] M. Pioro-Ladrière et al., Appl. Phys. Lett. 90, 024105 (2007). [19] J. M. Park et al., in 2015 IEEE Int. Elec. Dev. Meet. (IEDM) (IEEE, New York, 2015), pp. 26.5.1-26.5.4 Shuttling gates (blue) Source  gate Pulsed gates (red) DC: source  local demultiplexer  gate AC: source  gate Sensing dot plunger (purple) DC: source  local demultiplexer  gate AC: source  global demultiplexer  gate Drain contacts Measurement device  ohmic Table 1. Summary of signal routing for the four different type of control lines in the array design. Fig. 1: Image of a set of gate electrodes from a state-of-the-art device of electrostatically defined quantum dots. Fig. 2 Overview of the qubit architecture with a schematic breakdown of its components as described in the main text, including (a) the die containing (b) the quantum plane area, highlighting a single (c) module which contains a set of (d) unit cells. Qubits are color coded to distinguish data qubits (blue) and ancilla qubits (red), as defined in the surface code. Fig. 3 (a) Schematic of a unit cell containing four spin", + "die containing (b) the quantum plane area, highlighting a single (c) module which contains a set of (d) unit cells. Qubits are color coded to distinguish data qubits (blue) and ancilla qubits (red), as defined in the surface code. Fig. 3 (a) Schematic of a unit cell containing four spin qubits (green), operation regions (purple/orange), connected via shuttling channels (grey lines). (b) Qubit idling region. Four barrier gates (red) define the confinement potential and allow qubits into the shuttling channels (blue). Cyan circles represent vias. (c) Qubit operations region including control gates (red), sensing dot plunger (purple), source (S)/drain (D) ohmics (squares) and micromagnets (orange rectangles). (d) Two-qubit operation only regions. Fig. 4 (a) Schematic of a unit cell with locally integrated classical electronics. The color coding represents the same components in all the panels. (b) Circuit schematic of the components in (a), with the functionality described in the main text. fDAC (sDAC) are voltage sources for pulsed signals (DC biasing). Dashed red line denotes the quantum plane boundary in this and following panels. (c) Input/output schematic of the demultiplexer. (d) Schematic of a module. Demultiplexers are sequentially enabled by crossbar addressing controlled by multiplexers (orange blocks). (e) Zoom into the area surrounding a single unit cell in (d). (f) Schematic of the array of modules completing the quantum plane.", + "arXiv:cond-mat/0107042v1 [cond-mat.str-el] 2 Jul 2001 From Mott insulator to overdoped superconductor: Evolution of the electronic structure of cuprates studied by ARPES A. Damascelli 1, D.H. Lu, Z.-X. Shen Department of Physics, Applied Physics and Stanford Synchrotron Radiation Laboratory Stanford University, Stanford, CA 94305, USA Abstract We review our angle-resolved photoemission spectroscopy (ARPES) results on different layered oxide superconduc- tors, and on their insulating parent compounds. The low energy excitations are discussed with emphasis on some of the most recent issues, such as the Fermi surface and remnant Fermi surface, the pseudogap and d-wave-like dispersion, and lastly the signatures in the ARPES spectra of multiple electronic components, many body effects, and the superfluid density. We will focus on systematic changes in the electronic structure which may be relevant to the development of a comprehensive picture for the evolution from Mott insulators to overdoped superconductors. Keywords: Photoemission; ARPES; Electronic structure; Cuprates; Mott insulators; High temperature superconductors 1. Introduction Following their remarkable discovery in 1986 [1,2], high-temperature superconductors (HTSCs) have attracted great interest due to their scientific significance and enormous potential for applica- tions. The latter is obviously related to the high transition temperature shown by these compounds (Tc can be as high as 134 K in HgBa2Ca2Cu3O8−δ at atmospheric pressure [3]). Their scientific im- portance stems from the fact that the HTSCs high- light a major intellectual crisis in the quantum- theory of solids which, in the form of one-electron band theory, has been very successful in describing good metals and semiconductors (like Cu and Si, respectively) but has proven to be inadequate for 1 Corresponding author. E-mail: damascel@stanford.edu strongly correlated electron systems. The failure of the single-particle picture and the main con- ceptual issues involved in the study of HTSCs can be best illustrated starting from the phenomeno- logical phase diagram of n and p-type HTSCs [represented by Nd2−xCexCuO4 (NCCO) and La2−xSrxCuO4 (LSCO), respectively, in Fig. 1]. The T 2 dependence of the resistivity observed in the overdoped metallic regime is taken as evi- dence for Fermi liquid (FL) behavior. On the other hand, the applicability of FL theory (which de- scribes electronic excitations in terms of a weakly interacting gas of quasiparticles) to the ‘normal’ metallic state of HTSCs is questionable, because many properties do not follow canonical FL behav- ior. Most dramatic becomes the breakdown of FL theory and of the single particle picture upon ap- J. Electron Spectr. Relat. Phenom. 117-118, 165 (2001) Fig. 1. Phase diagram of n and p-type superconductors. proaching the undoped line of the phase diagram (x=0), where we find the antiferromagnetic (AF) Mott insulator. Mott-Hubbard insulators, because of the odd number of electrons per unit cell, are systems erroneously predicted by band theory to be paramagnetic metals (with a partially filled d- band in the case of transition metal oxides) [4– 6]. The reason for this failure lies in the on-site electron-electron repulsion U which is much larger than the bandwidth W. As a consequence, charge fluctuations are suppressed and these compounds are rather good insulators at all temperatures, with an optical gap U of", + "case of transition metal oxides) [4– 6]. The reason for this failure lies in the on-site electron-electron repulsion U which is much larger than the bandwidth W. As a consequence, charge fluctuations are suppressed and these compounds are rather good insulators at all temperatures, with an optical gap U of a few eV between the lower and upper Hubbard bands (LHB and UHB). As a mat- ter of fact, in the cuprates the Cu-O charge-transfer energy ∆is smaller than the on-site Coulomb re- pulsion U (see Fig. 7a), which characterizes these compounds more precisely as charge-transfer in- sulators [7]. However, because the first electron- removal state corresponds to the O-derived Zhang- Rice singlet band (ZRB), the cuprates are thought to be equivalent to an effective Mott-Hubbard sys- tem with the ZRB playing the role of the LHB, and an in-plane Cu-derived band as the UHB [8]. These states are separated by an effective Mott gap ∼∆. Therefore, many aspects of the physics of the cuprates are believed to be captured by the single- band Hubbard model [9]. This contains a kinetic- energy term proportional to the nearest neighbor (nn) hopping amplitude t, and an ad hoc Hubbard U term which accounts for electronic correlations: large Coulomb repulsion U favors electron local- ization and results in ‘frustration’ of the kinetic energy. In the strong coupling limit (U ≫t) at half filling (x = 0, i.e., one electron per Cu site), the AF state [10] results from the fact that, when nn spins are antiparallel to each other, the electrons gain kinetic energy by undergoing virtual hopping to the neighboring sites (because of the Pauli prin- ciple hopping is forbidden for parallel spins). By projecting out the doubly occupied states at large U [11], the low-lying excitations of the 1/2-filled Hubbard model are described by the t-J Hamil- tonian (J ∼t2/U). Away from half filling, the t-J model describes the so called ‘doped AF’, i.e., a system of interacting spins and mobile holes. The latter acquire a ‘magnetic dressing’ because they are perturbing the correlations of the spin back- ground that they move through. The challenge in the investigation of the elec- tronic properties of the cuprates is to sort out the basic phenomenology that can test the relevance of many-body models in describing the low lying excitations, in both the insulator and the doped metal. At the same time, it is also prudent to con- sider the influence of other degrees of freedom on the physical properties of these complex materials. For instance, phonon modes and lattice distortion can play a very important role, in particular when they couple to potential instabilities of the charge and/or spin degrees of freedom. Nevertheless, in order to address the scope of the current approach in the quantum theory of solids and the validity of the proposed models, a detailed comparison with experiments that probe the electronic properties and the nature of the elementary excitations is re- quired. In this context, ARPES has played a major role because it is the most direct method of study- ing", + "quantum theory of solids and the validity of the proposed models, a detailed comparison with experiments that probe the electronic properties and the nature of the elementary excitations is re- quired. In this context, ARPES has played a major role because it is the most direct method of study- ing the electronic structure of solids [12]. In this paper, we review our recent ARPES results on the cuprates. As the doping evolution allows a critical comparison between theory and experiment, we will discuss ARPES data on the HTSCs and their insulating parent compounds, focusing on systematic changes in the electronic structure that may be relevant to the development of a comprehensive picture for the evolution from Mott insulators to overdoped superconductors. 2 2. State-of-the-art ARPES In the early stages of the HTSC field, ARPES proved to be very successful in measuring the nor- mal state Fermi surface (FS), the superconduct- ing gap, and the symmetry of the order parameter [12]. During the past decade, a great deal of effort has been invested in further improving this tech- nique which now allows for energy and momentum resolution of, respectively, a few meV and ∼1% of the typical Brillouin zone (BZ) of HTSCs, thus ushering in a new era in electron spectroscopy and allowing a very detailed comparison between the- ory and experiment. To illustrate the capability of state-of-the-art ARPES, the novel superconduc- tor Sr2RuO4 is a particularly good example be- cause of its complex electronic structure and espe- cially because controversy has plagued the investi- gation of its FS topology. In addition, contrary to the cuprate HTSCs, this material can also be in- vestigated with other techniques, like de Haas-van Alphen (dHvA) experiments, thus providing a di- Fig. 2. ARPES spectra from Sr2RuO4 along the high sym- metry lines Γ-M, M-X, and Γ-X, as shown in the sketch depicting 1/4 of the 2D projected zone. Data from [17]. Fig. 3. (a) LEED pattern obtained with 450 eV electrons [17]. The arrows indicate superlattice reflections due to √ 2× √ 2 surface reconstruction. (b) EF intensity map [17]. Primary α, β and γ sheets of FS are marked by white lines, and replica due to surface reconstruction by black lines. rect comparison with the ARPES results. Whereas dHvA experiments [13], in agreement with LDA band-structure calculations [14], indi- cate two electron-like FSs β and γ centered at the Γ point, and a hole pocket α at the X point (as sketched in 1/4 of the projected zone in Fig. 2), early ARPES measurements suggested a different picture: one electron-like FS at the Γ point and two hole pockets at the X point [15]. The difference comes from the detection by ARPES of an intense, weakly dispersive feature at the M point just below EF , that was interpreted as an extended van Hove singularity (evHs). Although the evHs was ques- tioned in a later ARPES study [16], in which the feature detected at the M point was suggested to be a surface state (SS), a conclusive picture and a final agreement between ARPES data,", + ", that was interpreted as an extended van Hove singularity (evHs). Although the evHs was ques- tioned in a later ARPES study [16], in which the feature detected at the M point was suggested to be a surface state (SS), a conclusive picture and a final agreement between ARPES data, and dHvA and LDA results was reached only with the ‘new- generation’ of high-resolution photoemission data. Fig. 2 presents energy distribution curves (EDCs), along the high-symmetry directions of Sr2RuO4, recently reported by Damascelli et al. [17]. Owing to the high momentum and energy res- olution (1.5% of the BZ and 14 meV), we can now clearly identify several dispersive features crossing EF precisely where the α, β, and γ sheets of the FS are expected on the basis of LDA calculations and dHvA experiments (all detected features in Fig. 2 are labeled following their assignment). Around the M point we can also observe the sharp peak (labeled SS) that was initially associated with a hole-like sheet of FS centered at X [15]. 3 A Fermi energy intensity map can be obtained by integrating the EDCs over a narrow energy win- dow about EF (±10 meV). As the spectral function multiplied by the Fermi function reaches its maxi- mum at EF when a band crosses the Fermi energy, the FS is identified by the local maxima of the in- tensity map. Following this method, the α, β, and γ sheets of FS are clearly resolved, and are marked by white lines in Fig. 3b. In addition, we find some unexpected features: weak, yet well defined profiles marked by black lines. They can be recognized as a replica of the primary FS, and are related to the weak ‘shadow bands’ (SB) which show dispersion opposite to the primary peaks along Γ-M and Γ- X (see Fig. 2). The origin of the shadow bands as well as of the SS, can be identified with the intrin- sic instability of the cleaved surface of Sr2RuO4: inspection with LEED reveals superlattice reflec- tions corresponding to a √ 2 × √ 2 surface recon- struction (Fig. 3a), which is responsible for the folding of the primary electronic structure with re- spect to the M-M direction [17,18]. In light of these findings, the bulk FS determined by ARPES is con- sistent with the LDA and dHvA results. In addi- tion, ARPES provides essential information on the detailed shape of the α, β, and γ sheets of FS [17]. Fig. 4. Valence band dispersion for SCOC measured from the top of the band. Experimental data are taken from [19] (open circles), [20] (open triangles) and [21] (open squares). Dashed line: results from the t-J model [19]. Solid circles: self-consistent Born approximation (SCBA) for the t-t′-t′′-J model (t=0.35 eV, t′=−0.12 eV, t′′=0.08 eV and J=0.14 eV); solid lines are obtained by fitting the SCBA data [33]. The dotted line along the (π,0)-(0,π) direction represents the spinon dispersion given in [34]. 3. The Mott insulator The t-J model, briefly discussed in the introduc- tion, is of particular relevance to the low-energy features", + "t′′=0.08 eV and J=0.14 eV); solid lines are obtained by fitting the SCBA data [33]. The dotted line along the (π,0)-(0,π) direction represents the spinon dispersion given in [34]. 3. The Mott insulator The t-J model, briefly discussed in the introduc- tion, is of particular relevance to the low-energy features detected in ARPES on the cuprates. In fact, in ARPES experiments performed on the insulating parent compounds of the HTSCs, one photo-hole is injected in the CuO2 plane as a result of the photoemission process. Therefore, this experiment is the practical realization of a ‘single hole’ in the AF insulator, and the compar- ison of ARPES data and calculations based on the t-J model is particularly meaningful because the single-hole calculation is free from complica- tions such as charge ordering, which is difficult for small-cluster calculations to deal with. Experimental data taken from Ref. [19–21] for the energy dispersion of the quasiparticle (QP) peak in insulating Sr2CuO2Cl2 (SCOC) are shown in Fig. 4 (open symbols). Note that in the course of the paper we will use terms like QP in a loose sense for convenience, even though in most cases FL the- ory does not apply and well defined QP peaks can- not be identified in the ARPES spectra. The dis- persion along the (0,0)-(π,π) direction is charac- terized by a bandwidth W ≃0.3 eV. As pointed out in the first ARPES investigation on this com- pound [19], this result is in very good agreement with t-J model calculations [22] which show that, independent of the value of t, the dressing of the hole moving in the AF background reduces the QP bandwidth from 8t (∼3 eV for a free hole) to 2.2J (with J ≃125 meV in SCOC, as independently deduced from neutron scattering studies [23]). On the other hand, the t-J model also predicts a rela- tively flat dispersion [22] along the (π,0)-(0,π) di- rection (dashed line in Fig. 4), in contradiction to the more isotropic dispersion observed in ARPES around (π/2,π/2), with W ≃0.3 eV independent of the direction. Also the poorly defined lineshape and the spectral weight suppression observed at (π,0), which indicate the lack of integrity of the QP at those momenta, cannot be reproduced within the simple t-J model [21]. 4 Better agreement between the experimental dis- persion and the calculations (solid circles and solid line in Fig. 4) is obtained by adding second and third nn hopping (t′ and t′′, respectively) to the t-J Hamiltonian [21,24–33]. In fact, as t′ and t′′ describe hopping within the same magnetic sub- lattice, they do not alter the AF properties of the model at half filling; at the same time, they are not strongly renormalized by the AF correlations but contribute directly to the coherent motion of the hole and, therefore, have a substantial impact on the QP dispersion. The inclusion of these terms also helps in reproducing the suppression, as com- pared to (π/2,π/2), of the QP peak observed in ARPES at (π,0). However, nothing can be said about the line shape because the broadening is ar-", + "hole and, therefore, have a substantial impact on the QP dispersion. The inclusion of these terms also helps in reproducing the suppression, as com- pared to (π/2,π/2), of the QP peak observed in ARPES at (π,0). However, nothing can be said about the line shape because the broadening is ar- tificially introduced in the theory, which is a ma- jor limitation of this kind of approach. Most im- portantly, it can be shown that the suppression of the QP peak at (π,0) reflects a reduction of AF spin correlations: the additional hopping possibil- ities represented by t′ and t′′ induce a spin-liquid state around the photo-hole with momentum (π,0) [33]. As a consequence, one may expect to find in the ARPES results some signatures of spin-charge separation [9]. Within this context, it is interest- ing to note that the full QP dispersion observed for SCOC can be very well reproduced also by the spinon dispersion given in [34] [the dotted line in Fig. 4 shows the result along (π,0)-(0,π)]. In this case Laughlin argues in favor of the decay of the photo-hole injected in the 2D AF CuO2 plane into a spinon-holon pair [34], which is reminiscent of the flux phase physics [35–40], an extension of the early resonating valence bond (RVB) conjecture [9]. The discussion of the ARPES result on insulat- ing SCOC emphasizes a fundamental problem in the theoretical description of the doped 2D AF: the Heisenberg model is so strongly perturbed by the addition of mobile holes that, above a certain dop- ing level, some form of spin liquid may be a bet- ter ansatz than the long range ordered N´eel state. This point is centrally important to high-Tc su- perconductivity because HTSCs, which are poor conductors in the normal state, may be better re- garded as doped AFs, whose behavior differs fun- damentally from the FL paradigm. For this rea- son, the Bardeen-Cooper-Schrieffer (BCS) theory [41] which was developed for Fermi-liquid metals (i.e., weak electron correlations), and has been so successful in describing conventional superconduc- tors, does not have the appropriate foundation for HTSCs. A new approach may therefore be needed, and a necessary requirement for any theory aim- ing to capture the essential physics of high-Tc su- perconductivity must be the inclusion of the es- sential physics of the doped AF: the competition between AF and Coulomb interactions (which in- duce localization), and zero point kinetic energy (which favors delocalization). Along this direction, the most radical models seem to be those based on: (i) the RVB state and the related spin-charge separation picture [9,34–40,42–45], (ii) stripes [46– 57], and (iii) quantum criticality [58–63]. Indepen- dent of their details, these different theoretical ap- proaches have one important common denomina- tor: superconductivity is not caused by the pairing of two QPs, as in the BCS case, rather it is the pro- cess in which the QP itself forms. Furthermore, in the first two cases the driving mechanism for the superconducting phase transition is identified with the gain in kinetic energy, contrary to the standard theories of solids where any", + "two QPs, as in the BCS case, rather it is the pro- cess in which the QP itself forms. Furthermore, in the first two cases the driving mechanism for the superconducting phase transition is identified with the gain in kinetic energy, contrary to the standard theories of solids where any phase transition into a long-range ordered state is driven by the gain in potential energy. In the stripe or RVB models the hopping of pairs of holes perturbs the AF spin background less than individual holes. However, it is only when charge fluctuations become phase co- herent that the frustration of the kinetic energy is released, and superconductivity sets in. 3.1. Remnant FS and d-wave-like dispersion We mentioned above that both the relatively isotropic dispersion at (π/2,π/2), and the suppres- sion of QP weight at (π,0), observed from ARPES on SCOC and Ca2CuO2Cl2 (CCOC, similar in many respects to SCOC [64]), cannot be explained with the nn hopping t-J model. Better agreement 5 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.0 0.5 1.0 0 100 200 300 400 k = (π,0) 17.5% Dy 35% Dy 10% Dy 0% Dy Dy-Bi2212 (b) Tc Ca2CuO2Cl2 (insulator) 0K 25K 65K 85K Intensity (arbitrary unit) Energy relative to EF or peak at (π/2,π/2) (ev) (a) High Energy Gap (meV) |cos(kxa) - cos(kya)| / 2 Ca2CuO2Cl2 Tc= 0K 35 % Dy Tc= 0K 17.5 % Dy Tc=25K 10 % Dy Tc=65K Fig. 5. (a) High energy pseudogap versus |cos kx−cos ky|/2 for CCOC, and Dy-Bi2212. (b) Doping dependence of the (π,0) ARPES spectra. Data taken from [64]. with the experiment is obtained by including longer range hopping terms in the model. In this way, it is possible to also reproduce the doping dependence of the QP band structure and, in par- ticular, of the (π,0) ARPES spectra [28]. These are shown for optimally doped Bi2Sr2CaCu2O8 (Bi2212), Dy-Bi2212 and CCOC [64] in Fig. 5b. Note that for CCOC the zero in energy does not correspond to EF but to the peak position at (π/2,π/2) which, because of the Mott gap, is lo- cated ∼700 meV below the chemical potential and corresponds to the top of the valence band. In underdoped samples the QP peak at (π,0) loses co- herence and shifts to higher BE, while at (π/2,π/2) spectral weight still reaches the Fermi level. The lack of a FS crossing along the (π,0)-(π,π) cut in the normal state of the underdoped regime indi- cates the opening of a pseudogap [64–67], which appears to be characterized by the same d-wave symmetry found for the superconducting gap [68]. Because the maximum size of the pseudogap is found at the EF crossing along the (π,0)-(π,π) direction, and the dispersion around (π,0) is very weak, the doping dependence of the pseudogap magnitude can be deduced from the energy posi- tion of the QP peak at (π,0), as shown in Fig. 5b. However, the t-J model, even in its more ex- tended form, cannot completely account for the strong momentum dependence of the ARPES spec- tra from the undoped insulator [69]. In particular,", + "can be deduced from the energy posi- tion of the QP peak at (π,0), as shown in Fig. 5b. However, the t-J model, even in its more ex- tended form, cannot completely account for the strong momentum dependence of the ARPES spec- tra from the undoped insulator [69]. In particular, even though a drop of intensity of the lowest energy peak along (0,0)-(π,π), after the AF BZ bound- ary, is predicted by the t-J model [69,70], it is not as sharp as experimentally observed in SCOC and CCOC [19,64]. This limitation of the t-J model comes from having projected out the doubly oc- cupied states originally contained in the Hubbard model: whereas the momentum occupation num- ber n(k) is a strongly varying function of k in the intermediate-U Hubbard model at half filling, it is trivially equal to 1/2 in the t-J model which, there- fore, cannot describe the anomalous distribution of spectral weight in the single particle spectral func- tion. This effect is accounted for by the complete large U-limit of the Hubbard model (i.e., without neglecting certain terms in the large U perturba- tion theory), as shown by Eskes and Eder [69]. For U →∞the spectra of large-U-Hubbard and t-J model will coincide, and n(k)→1/2; however, the convergence in U is very slow. On the basis of finite size cluster calculations within the full strong cou- pling model, Eskes and Eder reproduced the sharp drop of intensity observed in the ARPES spectra from SCOC at the AF BZ boundary (referred to, by the same authors, as a ‘pseudo-FS’ [69]). A detailed experimental characterization of the k-dependence of the ARPES spectral weight for the undoped insulator has been presented by Ronning et al. [64], on the basis of n(k) mapping obtained by integrating the EDCs from CCOC over an energy window larger than the bandwidth. From the location of the steepest drops in n(k), which successfully gives the FS for overdoped Bi2212, they could define a ‘remnant-FS’ (RFS) for CCOC (Fig. 6a and 6b, bottom), which closely follows the AF BZ boundary. Note that matrix element effects also influence the k-dependence of 6 Fig. 6. FS and RFS (bottom) defined by the analysis of n(k) for overdoped Bi2212, and insulating CCOC. Note that while the FS in Bi2212 is the isoenergetic contour located at EF , the RFS is away from EF (because of the presence of the Mott gap), and a large d-wave-like dispersion (300 meV) is found along its contour. The latter defines a d-wave ‘gap’ for the insulator (top left), similar to the d-wave pseudo gap observed in the underdoped regime (top right), and due to strong correlations which deform the isoenergetic FS of the overdoped metal (from [64]). the intensity and alter the profile of the RFS [71]; however the sharp drop at the AF BZ boundary observed at different photon energies, in the ex- periment, and in the numerical results appears to be a robust feature, despite minor uncertainties related to the photon energy dependence of the photoionization cross section [72,73]. It is important to realize", + "[71]; however the sharp drop at the AF BZ boundary observed at different photon energies, in the ex- periment, and in the numerical results appears to be a robust feature, despite minor uncertainties related to the photon energy dependence of the photoionization cross section [72,73]. It is important to realize that the RFS is not a real FS (the system has a Mott gap) but identifies the same locus of rapid intensity drop referred to as a pseudo-FS by Eskes and Eder [69]. In addition, it does not even correspond to an isoenergetic con- tour in the QP dispersion (Fig. 6), similarly to the FS determined in the underdoped regime. The rele- vance of this approach is that, once a RFS has been determined, it is also possible to identify a ‘gap’ along its contour (in addition to the Mott gap), and try to compare it to the high energy pseudogap of the underdoped systems. As reported by Ronning et al. [64], the ‘high energy’ pseudogap (given by the position of the broad peak indicated by arrows in Fig. 5b) shows d-wave symmetry not only in the underdoped systems but also in the undoped insu- lator. This is shown in Fig. 5a where the dispersion of the high energy pseudogap along the FS (RFS for CCOC) is plotted against the d-wave functional form (a fit for CCOC is also shown, whereas the other lines are only guides to the eye). Although their sizes vary, the superconducting gap, the pseu- dogap of the underdoped system, and the gap of the insulator have the same nontrivial d-wave form, suggesting a common origin [74]. This is consistent with the idea of one underlying symmetry princi- ple [i.e., SO(5)] that unifies the AF insulating state and the d-wave superconducting state [75]. 4. Evolution of the electronic structure The work on the undoped insulator provides a starting point to understand the doping evolution of the electronic structure of the cuprates. Upon doping the system, AF correlations are reduced and a metallic state appears. Eventually (i.e., in the optimum and overdoped regime), the AF state is destroyed and a large LDA-like FS appears [12], with a volume which scales as (1 −x), counting electrons (x is the concentration of doped holes for p-type HTSCs), as expected within the FL ap- proach. In this context, the first important ques- tion to answer concerns the way the low energy states emerge in the underdoped regime. For x ≪ 1, two alternative scenarios have been proposed (see Fig. 7 for hole doping): (i) the chemical po- tential µ is pinned inside the charge-transfer gap ∆, as ‘in-gap states’ are created [76] (Fig. 7b); (ii) the chemical potential moves downwards into the top of the valence band and states are transferred from the UHB to the LHB because of correlations [77] (Fig. 7c). Another relevant question is how the 7 U ∆ µ (a) (b) (c) µ µ Fig. 7. Doping of a charge-transfer insulator (a): µ is pinned inside the charge-transfer gap and states move towards the chemical potential (b); alternatively,", + "the UHB to the LHB because of correlations [77] (Fig. 7c). Another relevant question is how the 7 U ∆ µ (a) (b) (c) µ µ Fig. 7. Doping of a charge-transfer insulator (a): µ is pinned inside the charge-transfer gap and states move towards the chemical potential (b); alternatively, µ shifts to the top of the valence band and spectral weight is transferred because of correlations (c). Figure taken from [77]. low-lying states evolve from underdoped to over- doped regime, where FL behavior seems to recover [12]. To better organize the discussion, let us lay- out some relevant theoretical models. They can be classified as: (i) those that preserve the underly- ing crystalline symmetry; (ii) those that break this symmetry. Note that also those scenarios based on a dynamical breaking of symmetry should be taken into account because ARPES is sensitive to the latter, due to the relatively high excitation energy. The first models to be mentioned among (i) are the FL and band structure perspectives [78,79], which sever the connection to the undoped AF insulator by assuming that the screening in the doped metal is strong enough for the FL formalism to recover; in this case a well defined FS is expected (Fig. 8a), with a volume proportional to (1−x). An alternative scenario considers the breakdown of FL theory due to umklapp scattering [80]. As a conse- quence, in the underdoped region of the phase di- agram, the FS is truncated near the saddle points at (π,0) and (0,π) because of the opening of spin and charge gaps. This results in four disconnected arcs of FS centered at (±π/2,±π/2), as shown in Fig. 8b. In agreement with a generalized form of Luttinger’s theorem, the area defined by the four arcs and by the umklapp gapped FS (dashed lines in Fig. 8b) encloses the full electron density. Among the broken-symmetry models, we find the RVB/flux-phase approach [35–40] that pre- dicts a FS given by four hole-pockets close to (±π/2,±π/2), as in Fig. 8c, which continuously evolve into a large FS upon increasing the hole con- centration. This is in a very similar spirit to that of the spin-density wave picture which also assumes a dynamical breaking of symmetry [81]. Another model belonging to (ii) is the stripe picture [48], which yields a momentum-space distribution of low-lying excitations. These are represented by the black patches in Fig. 8d, where the results obtained for an equal number of vertical and hor- izontal domains of disordered stripes are qualita- tively sketched (in this case the physics, together with the superposition of domains, conspires to give the appearance of a large LDA-like FS). There is actually another meaningful way to dif- ferentiate the four models discussed above: those depicted in Fig. 8a, 8b, and 8c assume that the sys- tem is spatially uniform (as far as Fig. 8b is con- cerned, one could talk about phase separation be- tween insulating spin-liquid and metallic regions, but only in momentum space [80]). To the con- trary, the model in Fig. 8d assumes that the system is spatially", + "assume that the sys- tem is spatially uniform (as far as Fig. 8b is con- cerned, one could talk about phase separation be- tween insulating spin-liquid and metallic regions, but only in momentum space [80]). To the con- trary, the model in Fig. 8d assumes that the system is spatially non-uniform: the formation of stripes is defined as the segregation of charge carriers into (π,0) (0,π) (π,π) (a) Γ (b) (c) (d) Fig. 8. FS obtained for the CuO2 plane from (a) LDA [12], (b) truncation of a 2D FS due to umklapp scattering [80], (c) RVB/flux-phase approach [36], and (d) stripe model for vertical and horizontal domains of disordered stripes [48]. 8 Fig. 9. Doping dependence of the ARPES spectra for LSCO at (π,0) and (π/2,π/2). EDCs were normalized to the integrated intensity of the valence bands (E >−0.9 eV), and for (π/2,π/2) were multiplied by a factor of two. Data taken from [84]. 1D domain walls which separate AF spin domains in antiphase with each other, and here, in particu- lar, disordered stripes are considered [48,82]. Each of the above discussed pictures captures some aspects of the experimental reality. In the course of the paper, we will try to compare ARPES data from various systems with the results of these models, aiming to identify the scenario that has the best overlap with experimental observations. This will also help us to answer the question whether different materials would favor different scenarios, and to address the relevance of degrees of freedom other than the electronic ones (e.g., lattice degrees of freedom in the case of the stripe instability). 4.1. La2−xSrxCuO4 In order to study the doping evolution of the low-energy electronic properties over the full dop- ing range, in particular in the vicinity of the metal- insulator transition (MIT), the most suitable sys- tem is LSCO. The hole concentration in the CuO2 plane can be controlled and determined by the Sr content x, from the undoped insulator (x = 0) to the heavily overdoped metal (x ∼0.35). In addi- tion, LSCO has a simple crystal structure with a single CuO2 layer, and no complications due to superstructure and shadow bands, as in the case of Bi2212 [12]. Another interesting aspect is the suppression of Tc at x = 1/8 which, together with the incommensurate AF long-range order observed in inelastic neutron scattering [83], has been dis- cussed as evidence for fluctuating stripes in LSCO (similar AF order accompanied by charge order- ing has been interpreted as a realization of ‘static stripes’ in La1.48Nd0.4Sr0.12CuO4 [82]). Let us start from the low doping region, near the MIT boundary. Fig. 9 presents the ARPES spec- tra at (π,0) and (π/2,π/2) as a function of doping, reported by Ino et al. [84]. The data was recorded under identical experimental geometry so that the photoionization matrix elements are the same. For the insulating samples (x≤0.03), the data is char- acterized by a high binding energy (BE) feature [∼0.5 eV at (π/2,π/2), and ∼0.7 eV at (π,0)], and is consistent with what we have discussed in the last", + "under identical experimental geometry so that the photoionization matrix elements are the same. For the insulating samples (x≤0.03), the data is char- acterized by a high binding energy (BE) feature [∼0.5 eV at (π/2,π/2), and ∼0.7 eV at (π,0)], and is consistent with what we have discussed in the last section for insulating SCOC [19,21], albeit the features are now broader. The remarkable result is that for x = 0.05 two features can be identified in the EDC at (π,0): in addition to the high BE one, reminiscent of the ZR singlet band of the AF insu- lator, a second shoulder is observable close to EF . Upon further doping the system with holes, a sys- tematic transfer of spectral weight from the high- BE to the low-BE feature takes place, and a well- defined QP peak develops near optimal doping. On the other hand, the results obtained at (π/2,π/2) are very different: first of all, the data shows an overall suppression of weight as compared to (π,0) [the EDCs plotted in Fig. 9 for (π/2,π/2) have been multiplied by a factor of 2]; second, in the nodal region [i.e., along (0,0)-(π,π)], a QP peak is ob- servable only for x ≥0.15 [84]. As we will discuss later, with different experimental geometries more spectral weight is detected in the nodal region, but the overall trend of the doping dependence of the electronic structure is robust. The overall dispersion of the spectral features seen in LSCO and their doping dependence is sum- 9 marized by the plot of the second derivative (taken with respect to the energy) of the ARPES spectra presented in Fig. 10. Upon increasing the doping level, we can clearly observe the building of near- EF weight first at (π,0), and then at (π/2,π/2). Furthermore, the second derivative emphasizes the presence of the high-BE feature in the heavily un- derdoped samples (coexisting with the low BE one, at least for x=0.05), which has a ∼200 meV lower BE at (π/2,π/2) than at (π,0), in agreement with what is observed on the undoped insulator SCOC [19]. The ARPES results from LSCO, in particu- lar the presence of two electronic components and the fact that the low BE feature emerges first at (π,0) and then at (π/2,π/2), suggest that the ef- fects of doping on the electronic structure of the correlated insulator cannot be accounted for by a simple shift of the Fermi level in a rigid band model [84]. In fact, in the latter case the lowest excita- tions would first appear at (π/2,π/2). This argu- Fig. 10. Second derivatives of the ARPES spectra from LSCO over a broad doping range. Data taken from [84]. ment is in agreement with another observation by Ino et al. [85]: from the analysis of direct and in- verse angle-integrated photoemission spectra, they concluded that the chemical potential µ is pinned inside the charge-transfer gap for x ≤0.12, and starts shifting downwards only for x≥0.12. All the above results seem to indicate that in-gap states are created upon doping the insulator (Fig. 7b). Ino et", + "direct and in- verse angle-integrated photoemission spectra, they concluded that the chemical potential µ is pinned inside the charge-transfer gap for x ≤0.12, and starts shifting downwards only for x≥0.12. All the above results seem to indicate that in-gap states are created upon doping the insulator (Fig. 7b). Ino et al. argued that the ARPES results from LSCO may be understood within the stripe pic- ture [84]. This would explain the pinning of the chemical potential for x ≤1/8 as a consequence of the segregation of doped holes into metallic do- main walls, which corresponds to the appearance of in-gap states. Furthermore, the suppression of nodal intensity at EF would be a direct conse- quence of the vertical and horizontal orientation of the metallic stripes (domains are expected for this microscopic phase separation) [51]. Here the con- jecture is that charge fluctuations would be sup- pressed along directions crossing the stripes, and is supported by finite-size cluster calculations [51]. The sudden increase of EF weight for x≥1/8 in the nodal region may indicate that above this doping level the holes overflow from the saturated stripes. Concerning the relevancy of the stripe scenario to the ARPES data from HTSCs, more insights could come from the investigation of Nd-LSCO, a model compound for which the evidence of spin and charge stripe-ordering is the strongest [82]. High momentum resolution ARPES data on La1.28Nd0.6Sr0.12CuO4 were recently reported by Zhou et al. [86], and are shown in Fig. 11, where spectral weight maps obtained by integrating the EDCs over 100 and 500 meV energy windows below EF are presented (Fig. 11a and 11b, respec- tively). The data was symmetrized in accordance with the four fold symmetry of the BZ [note that the symmetry with respect to the (0,0)-(π,π) line is directly observed in the raw data taken, in the same experimental geometry, over a whole quad- rant of the BZ]. The low-energy spectral weight is mostly concentrated in a narrow region along the (0,0)-(π,0) direction, which is confined between lines crossing the axes at ± π/4 [86]. 10 Fig. 11. Intensity maps obtained by integrating the EDCs over 100 meV (a), and 500 meV (b) below EF . White lines in (b) enclose the high intensity region. Data from [86]. The high intensity area of the BZ in Fig. 11b, which is suggestive of almost perfectly nested 1D FS segments, is consistent with the 1/8 doping level of the system. Zhou et al. [86] interpret these re- sults as a signature of a 1D electronic structure related to the presence of static stripes. As indi- cated by neutron and x-ray experiments [82], at 1/8 doping the 1/4-filled charge stripes are sep- arated by AF domains with a periodicity of 4a (Fig. 12), where a is the lattice parameter. This pic- ture is also consistent with various theoretical cal- culations [50,53,55]. In particular, the preponder- ance of low lying excitations at (π,0), which is ob- servable in Fig. 11a, is consistent with calculations for disordered stripes like, e.g., those summarized in Fig. 8d [48,56]. Concerning the macroscopic ori-", + "parameter. This pic- ture is also consistent with various theoretical cal- culations [50,53,55]. In particular, the preponder- ance of low lying excitations at (π,0), which is ob- servable in Fig. 11a, is consistent with calculations for disordered stripes like, e.g., those summarized in Fig. 8d [48,56]. Concerning the macroscopic ori- entation of the stripes, two orthogonal domains are expected, as shown in Fig. 12a and 12b. For each domain the FS consists of straight lines perpendic- ular to the direction of the charge stripe itself, and intersecting the axes of the BZ at ± π/4 (Fig. 12c and 12d). The intensity distribution observed in Fig. 11b, would then result from the superposition of two perpendicular FSs reflecting the presence of orthogonal domains. The above interpretation of the ARPES data on Nd-LSCO would also provide a possible explana- tion for the origin of the two components seen in the ARPES spectra of LSCO near the MIT bound- ary (Fig. 9). In the static picture discussed above, the signal from the AF insulating regions would be pushed to high BE because of the Mott gap, whereas the charge stripes would be responsible for -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 kx ky kx 4a (a) (b) (c) (d) Fermi surface Fermi Surface Fig. 12. Horizontal (a) and vertical (b) static stripes, and their corresponding FS expected to be defined by the lines |kx|=π/4 (c), and |ky|=π/4 (d), respectively (from [86]). the component near EF . In this sense, the stripe interpretation is rather appealing. On the other hand, the picture discussed in [86] is based on the assumption of an extreme charge disproportiona- tion which is usually not found in charge ordering transitions, even in lower symmetry systems such as ladders [87]. Nevertheless, the qualitative pic- ture presented in Fig. 12 may capture the relevant physics even for a less extreme case of charge dis- proportionation. There are also some results which cannot be sat- isfactorily explained within the framework of static stripes. For example, in both LSCO (Fig. 10) and Nd-LSCO [86], the QP band along the (0,0)-(π,0) direction is characterized by a considerably fast dispersion, contrary to what is expected for an ideal 1D system which typically does not exhibit any dispersion perpendicularly to its main axis. Furthermore, matrix element effects have to be cautiously considered when interpreting ARPES data, especially in dealing with the integrated spec- tral weight. In fact, although the integration of the EDCs over a large energy window gives an es- timate for the momentum-dependent occupation number n(k) = R A(k, ω)f(ω)dω, the latter quan- 11 Fig. 13. ARPES spectra of underdoped (x=0.1) and heav- ily overdoped (x=0.3) LSCO, from [90]. Insets show mea- sured k-space points and incident light polarization. tity is weighted by the photoionization cross sec- tion and thus may contain extrinsic artifacts [88]. In order to gain more insight into this issue, in par- ticular in relation to the straight segments of FS observed in Nd-LSCO and to the suppression of", + "k-space points and incident light polarization. tity is weighted by the photoionization cross sec- tion and thus may contain extrinsic artifacts [88]. In order to gain more insight into this issue, in par- ticular in relation to the straight segments of FS observed in Nd-LSCO and to the suppression of the nodal state, Zhou et al. [89] extended the mea- surements to the second zone, and varied polariza- tion and orientation of the incoming electric field to enhance the spectral intensity in the (π/2,π/2) region. As a result, the presence of the nested seg- ments of FS near (π,0) and (0,π) was confirmed. On the other hand, appreciable spectral weight at the Fermi level was found in the nodal region [89], which appears to become more intense upon in- creasing the Sr concentration in both Nd-LSCO and LSCO, and is stronger in Nd-free LSCO for a given Sr content [89]. A possible way of under- standing these results within the stripe context, as suggested by Zhou et al. [89], is that the nodal state and the dispersion along (0,0)-(π,0) may arise from disorder or fluctuation of the stripes, where the holes leak into the AF regions [48]. As a mat- ter of fact, the detected FS [89], which is com- posed by straight patches at (π,0) and (0,π) con- nected by ‘nodal segments’, closely resembles the one depicted in Fig. 8d. Alternatively, this experi- mental FS may result from the coexistence of site- centered and bond-centered stripes [56]. Both of these scenarios suggest that the charge dispropor- tionation is not extreme, as also indicated by dy- namical mean-field calculations [54]. Lastly, it was noted that this composite FS has the appearance of a large LDA-like FS [89], like the one depicted in Fig. 8a. Because of disorder and the fact that the charge disproportionation is less than in the idealized stripe model of Fig. 12, one may still talk about the FS as the locus of low-lying excitations in k-space. This is particularly true for highly doped cases as the stripe effect should dis- appear. This existence of a FS in a striped sys- tem is found to be true in both cluster perturba- tion theory [56] and dynamical mean-filed type of calculations [48]. It is then meaningful to discuss the doping evolution of this LDA-like FS. ARPES spectra for underdoped (x = 0.1) and overdoped (x = 0.3) LSCO, reported by Ino et al. [90], are shown in Fig. 13. Note that the spectral features tend to be broad, which may be related to charge inhomogeneity. Nevertheless, some Fermi crossings are still observable in part of the BZ, especially in overdoped samples. For x = 0.1, along the di- rection (0,0)-(π,0)-(2π,0) at 29 eV photon energy (Fig. 13a), a broad QP peak emerges from the back- ground, disperses towards EF without crossing it, Fig. 14. FS of LSCO for x = 0.1 (a) and x = 0.3 (b), taken from [90]. Thick (thin) error bars denote FS crossings observed (folded by symmetry). For x = 0.1, FS crossings correspond to minimum-gap", + "peak emerges from the back- ground, disperses towards EF without crossing it, Fig. 14. FS of LSCO for x = 0.1 (a) and x = 0.3 (b), taken from [90]. Thick (thin) error bars denote FS crossings observed (folded by symmetry). For x = 0.1, FS crossings correspond to minimum-gap loci; as no dispersive feature is observed at (π/2,π/2) near EF (see EDCs in Fig. 13c), the dotted curve is drawn so that the FS area is ∼90% of the BZ area (in the respect of the Luttinger theorem). 12 and then pulls back in the second BZ. Similar re- sults are obtained at 22.4 eV (Fig. 13b), the only difference being a decrease of intensity in the first BZ due to matrix element effects specific to this photon energy [90]. Along (π,0)-(π,π) the QP peak (Fig. 13d), with maximum binding energy (BE) at (π,0), disperses almost up to EF , loses inten- sity, and disappears. The leading-edge midpoint never reaches EF because of the superconducting gap (∼8 meV) opening along the FS at this tem- perature. In this case, the underlying FS is iden- tified by the locus of the minimum gap [91], lo- cated here at (π,0.2π). Along the nodal direction no clear peak can be identified (Fig. 13c), as dis- cussed above. However, having detected a band be- low EF at (π,0), the authors conclude that for x= 0.1 the FS of LSCO is hole-like in character and centered at (π,π), as shown in Fig. 14a [90]. By comparing the EDCs from heavily overdoped and underdoped LSCO (Fig. 13e and 13a, respec- tively), we see a striking difference: for x=0.3 the QP peak present along this cut has almost disap- peared at (π,0). The decrease of intensity, together with a leading-edge midpoint now located above EF , provides evidence for the QP peak crossing EF just before (π,0). The FS thus determined for heavily overdoped LSCO (Fig. 14b) is electron-like in character and is centered at (0,0). Furthermore, careful investigations by Ino et al. [92] and Zhou et al. [89] show that the FS changes from hole-like to electron-like for x ≃0.15 - 0.2. In summary, what has emerged from the study of the LSCO system is a very complex and in- triguing picture, characterized by some contrast- ing aspects: neither a simple stripe model, nor any of other models proposed in Fig. 8 can provide a satisfactory explanation for the complete body of available data. As we have discussed, the stripe picture, when disorder/fluctuations and more real- istic charge disproportion are considered, has the advantage of qualitatively explaining the data over the entire doping range, including the presence of two electronic components, the straight FS seg- ments, and the lack of a chemical potential shift in the very underdoped regime. On the other hand, on a more quantitative level, there are still many open questions. On the experimental side, two is- sues should be carefully considered. The first one is the role of matrix element effects. In fact, ab initio calculations of matrix elements are still unavailable for", + "underdoped regime. On the other hand, on a more quantitative level, there are still many open questions. On the experimental side, two is- sues should be carefully considered. The first one is the role of matrix element effects. In fact, ab initio calculations of matrix elements are still unavailable for LSCO, and the tight binding fits do not repro- duce the results to a satisfactory degree. There- fore, the most robust information at present may come from the analysis of the systematic changes observed in data recorded under identical experi- mental conditions (e.g, as the doping dependence studies discussed in this section). Second, the sys- tem may consists of metallic stripes aligned along both [1,0] and [1,1], and characterized by smaller and larger charge disproportionation, respectively. If, on the one hand, this scenario might explain the photoemission data, on the other hand the coexis- tence of two phases is observed only near 5% dop- ing. On the theoretical side, it is unclear how the quasi-1D electronic structure of the stripe phase can be smoothly connected to the 2D electronic behavior of the overdoped regime. Although more effort has to be invested in the study of the elec- tronic properties of charge-ordered systems and, in particular, in investigating the role of the electron- lattice interaction, encouraging numerical studies were recently reported [54,56], which suggest that the spectral properties of this materials show a non trivial superposition of 2D AF and 1D metal- lic behavior. In particular, the dispersion observed in ARPES perpendicular to the stripe direction would stem from the AF domains [54,56]. 4.2. Bi2Sr2CaCu2O8+δ In proceeding with the comparative study of the cuprates, let us now turn our attention to Bi2212 which is the HTSC system most intensively investi- gated by ARPES (thanks to the presence of a nat- ural cleavage plane between the BiO layers). Due to sample quality issues, most of the Bi2212 exper- iments were carried out near optimal doping, and there is almost no information on the electronic structure near the MIT boundary. Here we con- 13 centrate on cases with a doping of 10% or higher. Therefore, we cannot answer the question whether the two-component electronic structure observed in the LSCO system with a doping of 5-7% is also present in Bi2212 cases. It is still an open question as to how the metallic state emerges in this system. Bi2212 can also be considered as the most de- bated HTSC system as far as ARPES is concerned, because of the complexity of the electronic struc- ture near (π,0). These complications arise from the detection of additional features besides those re- lated to the primary electronic structure: ‘shadow bands’ (possibly reflecting AF correlations [93] or the presence of two formula units per unit cell [12]), and ‘umklapp bands’ (due to the diffraction of the photoelectrons from the superstructure present in the BiO layers [94]). As a result, around (π,0) two main bands, two shadow bands, and four umklapp bands cross the Fermi level. One additional prob- lem with Bi2212 is that there are no reliable band calculations: all", + "(due to the diffraction of the photoelectrons from the superstructure present in the BiO layers [94]). As a result, around (π,0) two main bands, two shadow bands, and four umklapp bands cross the Fermi level. One additional prob- lem with Bi2212 is that there are no reliable band calculations: all theoretical results predicted a BiO FS that has never been observed [12]. In the fol- lowing, we will discuss the current understanding of these and other more recent issues of the inves- tigation of Bi2212 and its electronic properties. 4.2.1. Fermi surface After an initial debate, a consensus has been reached concerning the absence of the BiO pocket, which was predicted for Bi2212 by band structure calculations. Furthermore, the general consensus is in favor of a hole-like FS centered at (π,π), with a volume consistent with the electron den- sity in accordance with the Luttinger theorem [12,93,94]. In contrast to an earlier study that suggested the presence of an electron-like FS due to bilayer splitting [95], it was argued that there is no conclusive evidence for this effect [94]. For a period of time, the hole-like FS was believed to be the only FS feature in Bi2212 [96–99] over the doping range going from underdoped to over- doped samples (Tc ∼15 K and 67 K, respectively). These conclusions seem to be in contrast with the case of LSCO, where a crossover from a hole to Fig. 15. FS given by the integrated weight at EF for opti- mally (a) and Pb-doped (b) Bi2212 (from [107] and [108]). electron-like FS is clearly evident near optimal doping [90,92]. Recently, other reports questioned this picture, arguing that one simple hole-like FS may not be a complete characterization of the low-lying excitations in Bi2212 [100–102]. These studies suggested an electron-like FS centered at the Γ-point [101,102] or, possibly, two co-existing electronic components, resulting in electron and hole-like FSs [100]. These suggestions were op- posed by other groups which claimed that only one hole-like FS is supported by the ARPES data once the effects of the photon energy dependence of the matrix elements in the (π,0) region are taken into account [103–106]. To better illustrate the problems discussed above, we show in Fig. 15a the FS of optimally doped Bi2212 determined by integrating over a 7 meV window about EF the EDCs taken with unpolarized HeI radiation (from [107]). The data have been measured on a whole quadrant of the BZ to verify the symmetry between (π,0) and (0,π), and then have been symmetrized with respect to the (0,0)-(π,π) line to compensate for the differ- ent data sampling along horizontal and vertical directions. Inspection of the EDCs shows a Fermi crossing in going from (π,0) to (π,π), although the band is very flat [95], and from (0,0) to (π,π), in agreement with the picture of a hole-like FS cen- tered at (π,π). The latter is observable in the EF intensity map of Fig. 15a together with the two ghost FSs due to the umklapp bands and located on each side of the primary FS. In Fig. 15a", + "to (π,π), in agreement with the picture of a hole-like FS cen- tered at (π,π). The latter is observable in the EF intensity map of Fig. 15a together with the two ghost FSs due to the umklapp bands and located on each side of the primary FS. In Fig. 15a we can also observe a considerable amount of weight around (π,0). Because of the complexity of the 14 Fig. 16. n(k) plot (a), and EF intensity map (b) obtained by integrating the EDCs over 580 meV and 7 meV below EF , respectively. Dotted blue lines indicate the conventional hole-like FS; solid red lines the FS defined as the locus of the leading-edge midpoints in the EDCs (from [109]). EDCs in this region and the absence of a clear drop in intensity along (0,0)-(π,0), one cannot make strong statements concerning an electron-like FS. In relation to a possible electron-like topology of the Bi2212 FS in the overdoped regime, inter- esting results have been very recently reported by Bogdanov et al. [108,109]. The authors performed a detailed study of the FS topology in slightly over- doped Pb-Bi2212 (Tc=82.5 K). The advantage of working on Pb-doped Bi2212 is that Pb substi- tutes into the BiO planes, suppressing the super- structure and therefore the complications related to the umklapp bands at (π,0). As shown by the EF mapping presented in Fig. 15b [108], this system represents a strong case for an electron-like FS: in going from (0,0) to (π,0), a decrease of intensity, corresponding to a Fermi crossing, is now clearly observed near (π,0). However, as earlier data on the Pb-doped system by Borisenko et al. [104] were interpreted as evidence for a hole-like FS in Pb- Bi2212 also, and for the universality of the hole-like FS topology in Bi2212, one must examine these re- sults in greater detail. Bogdanov et al. [109], used different methods and photon energies to deter- mine the FS in different BZs (Fig. 16). More than 4000 EDC were collected and normalized by the integrated intensity from a 100 meV window above EF (i.e., energy and k-independent background), to compensate for variation of the photon flux and for the non-uniform response of the detector. The first indication for an electron-like FS was obtained from the analysis of the leading-edge midpoint in the EDCs. The results (independent of the zone) are plotted as solid red lines in Fig. 16. This topol- ogy is confirmed by the n(k) plot and EF inten- sity map (Fig. 16a and 16b, respectively): both the maxima in the EF map and the steepest inten- sity drops in the n(k) plot overlap with the EDC- derived FS (red lines). Note that the intensity is strongly suppressed in the first zone because of matrix element effects. However, inspection of the EDCs shows that all the features detected in the second BZ are present also in the first, although significantly weaker. As Pb doping does not modify the CuO2 layer, the FS measured on Bi2212 and Pb-Bi2212 is in- dicative of the electronic structure of the same CuO2 plane,", + "of the EDCs shows that all the features detected in the second BZ are present also in the first, although significantly weaker. As Pb doping does not modify the CuO2 layer, the FS measured on Bi2212 and Pb-Bi2212 is in- dicative of the electronic structure of the same CuO2 plane, and should show similar dependence upon hole doping. While the results reported by Bogdanov et al. [108,109] do not rule out the co- existence of a hole-like FS sheet, which may be apparent under different experimental conditions [103–106] as shown, for example, in Fig. 15a [107], they suggest that the scenario of a hole-like FS in Bi2212 over the whole doping range is incomplete, and identify a possible similarity between LSCO and Bi2212 along these lines. Due to the limited doping range investigated, the ARPES results from Bi2212 are insufficient to conclude in favor of any of the four scenarios summarized in Fig. 8. While the FS of optimally doped Bi2212 resembles the one reported in Fig. 8a and 8d, in the underdoped region, due to the open- ing of the pseudogap along the underlying FS (see next section), the data is reminiscent of the mod- els depicted in Fig. 8b and 8c. The distinction be- tween Fig. 8a and 8d has to be determined on the basis of the behavior near (π,0). This issue is prob- ably relevant to the controversy over the FS in the same k-space region, and is important in connec- tion to the possible signature of the superfluid den- sity in the ARPES data, as we will elaborate later. The distinction between the models described in Fig. 8b and 8c lies in the detection of the ‘shadow FS’ given by the dashed line in Fig. 8c. Although it has been argued that the case of Fig. 8c does not apply to Bi2212 [96], in contrast to an earlier report 15 [93], no consensus has been reached because, first, the shadow FS may be simply too weak to be seen and, second, this momentum region is complicated by the presence of shadow and umklapp bands. 4.2.2. Pseudogap Despite the controversy over the topology of the FS, many of the ARPES results are still reliable and the same holds for the qualitative descrip- tions that were developed on the basis of those results. This is particularly true with regard to a doping dependent study performed under identi- cal experimental conditions. One of these impor- tant results is the normal state excitation gap or pseudogap [65–67]. This feature was first recog- nized in the photoemission spectra in attempting to connect the Bi2212 data to that from the in- sulator [110,65,66]. An example of the detection of the pseudogap in ARPES spectra is presented in Fig. 5b where, following the idea of Laughlin [34], we compare the data at (π,0) as a function of Dy concentration. As the Dy content is increased and the system enters the underdoped regime, the ‘hump’ shifts to high BE, reflecting the opening of a normal state gap. As the size of the pseudogap is rather", + "of Laughlin [34], we compare the data at (π,0) as a function of Dy concentration. As the Dy content is increased and the system enters the underdoped regime, the ‘hump’ shifts to high BE, reflecting the opening of a normal state gap. As the size of the pseudogap is rather large and the band dispersion near (π,0) is weak, the detection of the pseudogap is relatively insensitive to the FS topology: either FS topol- ogy will lead to qualitatively similar conclusions. Hence, the normal state gap and its doping depen- dence are robust features in the ARPES spectra. The main characteristics of the pseudogap can be summarized as follows [111]: (i) the effect is strong in underdoped samples, persists to optimal doping, and disappears in overdoped samples [65– 67]. (ii) The gap has two energy scales, with the low energy one being identified by the location of the leading-edge midpoint, and the higher energy one by the position of the hump. The lower energy scale has a d-wave like momentum dependence, similar to the one of the superconducting gap, but with a gapless arc near the nodal region. The doping de- pendence of the two energy scales track each other [66,97,98,112,113]. (iii) Upon decreasing the hole Fig. 17. (a) T-dependent ARPES spectra from overdoped Bi2212 (Tc = 84 K). Data was collected around (π,0) in momentum space (shaded area in the sketch of 1/4-BZ). Inset: enlarged view of EDCs taken just above Tc. (b) Dop- ing dependence of the superconducting-state (π,0) spectra of Bi2212, for T ≪Tc. From [117] and references therein. concentration, the size of the leading-edge pseudo- gap increases, in contrast to the decreasing of Tc. This provides an important piece of evidence for non-BCS behavior of the superconducting transi- tion in the underdoped regime. 4.2.3. Signature of superfluid density We will now turn our attention to the well-known peak-dip-hump feature [12,113–116] detected be- low Tc in Bi2212, in the region around (π,0). In particular, we will discuss the detailed investiga- tion of doping and temperature dependence of this feature, which has been very recently reported by Feng et al. [117]. ARPES spectra from Bi2212 were collected at many different temperatures in the re- gion around (π,0), as sketched in Fig. 17a. Note that this average in momentum space, which re- sults in a considerably improved signal-to-noise ra- tio, does not appreciably alter the k-information because the QP dispersion is weak in this region. In Fig. 17a, where EDCs from overdoped Bi2212 with Tc=84 K (thus labeled as OD84) are displayed, we can see the typical peak-dip-hump structure (open triangle, cross, closed triangle, in Fig. 17a). 16 It becomes more pronounced upon reducing the temperature below Tc but it is still visible slightly above Tc, as shown in the inset of Fig. 17a. The re- sults obtained at low temperatures (∼10 K) for dif- ferent doping levels are displayed in Fig. 17b (UD for underdoped, OP for optimally doped, and OD for overdoped). The peak, not observed in the very underdoped samples, grows with doping and de- creases again slightly", + "of Fig. 17a. The re- sults obtained at low temperatures (∼10 K) for dif- ferent doping levels are displayed in Fig. 17b (UD for underdoped, OP for optimally doped, and OD for overdoped). The peak, not observed in the very underdoped samples, grows with doping and de- creases again slightly after optimal doping. Feng et al. [117], through a phenomenological fit- ting procedure, were able to quantify the evolution of the peak intensity. In order to extract meaning- ful and reliable information (i.e., independent of artifacts due to k-dependence of matrix elements and/or different experimental conditions for the different samples), the authors focused on the ratio between the relative intensity of the peak and the total spectrum intensity (integrated from -0.5 to 0.1eV). The temperature and doping dependence of this quantity, referred to as the ‘superconduct- ing peak ratio’ (SPR), are presented in Fig. 18a and 18d. From the comparison (Fig. 18) with many superfluid-related quantities measured in Bi2212, YBa2Cu3O7−δ, and LSCO, two major conclusions can be drawn: (i) the remarkable similarity of the data presented in Fig. 18 strongly suggests a uni- versality in the superconducting properties of the cuprates; (ii) ARPES, which mainly probes single- particle excitations of the condensate and therefore directly measures the strength of the pairing (i.e., superconducting gap), can also provide informa- tion on the phase coherence of the superconducting state (usually inferred from those techniques which directly probe the collective motion of the conden- sate). This latter point is shown, as a function of hole concentration, by the remarkable resemblance of the SPR to the superfluid density (ns) measured by µSR (Fig. 18b), the condensation energy U from the specific heat, and the jump in the specific heat coefficient (Fig. 18c). In addition, upon increasing temperature the SPR decreases in a way similar to ns, as measured by microwave and µSR spec- troscopy (Fig. 18d and 18e), with an abrupt drop near Tc (disappearance of the phase coherence), rather than at T ∗(opening of the pseudo-gap in Fig. 18. Doping dependence, for T 1 for some direction α) vanishes as 1/d. Furthermore, the considered vectors are of minimal Euclidean length |τ| hinting at maximal overlap, i.e., largest |tτ| for fixed taxi-cab distance ||τ|| and fixed D. By deriving a recursion relation for ǫD(k) we have established that ǫ(k) = ∞ X D=1 t∗ D √ D! HeD(ǫhc k ) =: F(ǫhc k ) . (7) Using the orthogonality of the Hermite polynomials, one may express the hopping matrix elements in terms of the transformation function", + "D. By deriving a recursion relation for ǫD(k) we have established that ǫ(k) = ∞ X D=1 t∗ D √ D! HeD(ǫhc k ) =: F(ǫhc k ) . (7) Using the orthogonality of the Hermite polynomials, one may express the hopping matrix elements in terms of the transformation function F(x): t∗ D = 1 √ 2πD! Z ∞ −∞ dǫ F(ǫ) HeD(ǫ) e−ǫ2/2 . (8) 4 Specializing on the case of a monotonic transformation function F(x) (with derivative F′(x)), we can write ρ(ǫ) = 1 F′(F−1(ǫ)) ρhc(F−1(ǫ)) (9) which leads to F−1(ǫ) = √ 2 erf−1 \u0012 2 Z ǫ −∞ dǫ′ρ(ǫ′) −1 \u0013 . (10) Furthermore, the Fermi velocity vk = ∇ǫk can be computed: vk = F′(F−1(ǫ))vhc k = ρhc( √ 2 erf−1(2 R ǫ −∞dǫ′ρ(ǫ′) −1)) ρ(ǫ) vhc k . (11) A practical application of the general formalism proceeds as follows: 1. compute F−1(ǫ) from arbitrary target DOS ρ(ǫ) using (10) 2. invert function (numerically or analytically) to obtain F(ǫ) 3. evaluate transport properties, e.g., ⟨|vk|2⟩(ǫ) or ˜ρ(ǫ) using (11) 4. optionally determine microscopic model parameters t∗ D using (8) The only choice inherent in this procedure beyond the usual assumptions for large dimensions is contained in step 1 which by construction produces a monotonic transformation function F. The optical f-sum rule reads Z ∞ 0 dω σxx(ω) = σ0 4d ⟨˜ρ′(ǫ) ρ(ǫ) ⟩= σ0 4d Dh F′′(F−1(ǫ)) −F−1(ǫ)F′(F−1(ǫ)) iE . (12) Here, the first equality follows from (2), i.e., is generally valid within the DMFT while the second expression in terms of the transformation is specific to the formalism developed within this section. Example: Flat-band System One interesting limiting case of a monotonic transformation function which can be treated analytically is the step func- tion F(x) = 2Θ(x) −1 corresponding to a flat band DOS of the form ρ(ǫ) = (δ(ǫ−1) + δ(ǫ+1))/2. For this case, the hopping matrix elements read (trivially, t∗ 2n = 0): t∗ 2n+1 = (−1)n r 2 π (2n −1)!! p (2n + 1)! n→∞ −→ (−1)n(πn)−3/4 . (13) The asymptotic exponent −3/4 is only slightly smaller than the threshold value −1/2 required for a finite variance R ∞ −∞dǫ ǫ2ρ(ǫ) = P D t∗ D 2. For a rectangular model DOS, t∗ D ∼2−nn−3/4 already decays exponentially fast. 5 a) b) 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 <|vk|2> ε hypercubic / stacked Millis / Freericks this work 0 0.1 0.2 0.3 0.4 0.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 ~ρ(ε) ε 5th NN hopping 3rd NN hopping exact d=5 d=6 d=100 Figure 2. a) The average squared Fermi velocity ⟨|vk|2⟩(ǫ) is constant for the hypercubic lattice (or for the x-component of a stacked lattice); in contrast it vanishes for the isotropic lattice defined in this work. For comparison, the form suggested by Millis is also shown. b) Resulting function ˜ρ(ǫ) of the full isotropic model (solid line) in comparison with truncated models (Dmax = 3 or Dmax = 5), evaluated in finite dimensions 5 ≤d ≤100. 3. Application to the", + "isotropic lattice defined in this work. For comparison, the form suggested by Millis is also shown. b) Resulting function ˜ρ(ǫ) of the full isotropic model (solid line) in comparison with truncated models (Dmax = 3 or Dmax = 5), evaluated in finite dimensions 5 ≤d ≤100. 3. Application to the “Bethe” semi-elliptic DOS In this section, we will apply the new formalism to the Bethe semi-elliptic DOS in order to determine a corresponding tight-binding Hamiltonian de- fined on the hypercubic lattice with the same local properties as the Bethe lattice (with NN hopping) in the limit d →∞. From (11), we derive the average squared Fermi velocity defined in (3) in closed form: ⟨|vk|2⟩(ǫ) = 2π 4 −ǫ2 exp  −2 erf−1\u0010ǫ p 1 −ǫ2/4 + 2 arcsin(ǫ/2) π \u0011!2 . (14) Here, we have used the fact that ⟨|vk|2⟩(ǫ) is effectively constant (and equals 1 for unit variance and lattice spacing) in the hypercubic case. The result (solid line in Fig. 2a) has all the qualitative features expected for this observable in any finite dimension: ⟨|vk|2⟩(ǫ) is maximal near the band center, strongly reduced for large (absolute) energies and vanishes at the band edges: states at a (noninteracting) band edge do not contribute to transport. The violation of this principle in the stacked case (dashed lines in Fig. 2a), which corresponds to an application of the hc formalism to the Bethe DOS with ⟨|vk|2⟩constant up to the band edges, is clearly pathological. Therefore, our method has not only the merit of yielding isotropic transport, but also of avoiding unphysical behavior. In order to determine the microscopic model, we have to apply (8) to the numerically evaluated transformation function F. Again, the scaled hopping matrix elements fall offexponentially fast: only a fraction 10−3 6 a) b) 0 0.1 0.2 0.3 0 1 2 3 4 5 A(ω) |ω| U=4.0 U=4.6 U=5.0 U=5.5 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 σ(ω) |ω| U=4.0 U=4.6 U=5.0 U=5.5 0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 6 7 8 ∫0 ω dω’ σ(ω’) ω Figure 3. Numerical results for the half-filled Hubbard model with semi-elliptic DOS for d →∞in the paramagnetic phase at T = 0.05. a) Local spectral function A(ω) obtained from QMC (using a discretization ∆τ = 0.1) and MEM. b) Optical conductivity σ(ω) for the isotropic “redefined Bethe lattice”. The inset shows the partial f-sum R ω 0 dω′σ(ω′). of the total energy variance arises from hopping amplitudes beyond third nearest neighbors and only a fraction 10−6 results from hopping beyond 9th- nearest neighbors. This result suggests that properties of the model should be robust with respect to truncation. In fact, ˜ρ(ǫ) (and consequently the definition of σ(ω)) hardly changes when hopping is cut offbeyond 3rd or 5th nearest neighbors, even when evaluated in finite dimensions as seen in Fig. 2b. This behavior is very general so that results for σ(ω) of a local theory in finite dimensions will depend on d predominantly via the interacting DOS A(ω) and only very little via", + "cut offbeyond 3rd or 5th nearest neighbors, even when evaluated in finite dimensions as seen in Fig. 2b. This behavior is very general so that results for σ(ω) of a local theory in finite dimensions will depend on d predominantly via the interacting DOS A(ω) and only very little via ˜ρ(ǫ). The local spectral functions for T = 0.05, i.e., slightly below the critical temperature T ∗≈0.055, are shown in Fig. 3a as obtained from QMC/MEM [5]. In the metallic phase, the spectral density at the Fermi level (ω = 0) is approximately pinned at the noninteracting value ρ(0) = 1/π ≈0.32 for U ≲4.4. The quasiparticle weight decreases drastically and a shoulder develops for U ≳4.6 before a gap opens for U ≳4.8. An application of (2) to these spectra for the isotropic model characterized by (14) yields the estimates for the optical conductivity σ(ω) shown in Fig. 3b. A low- frequency Drude peak (of Lorentzian form) and a mid-infrared peak at ω ≈U/2 are present in the metallic phase and decay towards the metal- insulator transition at U ≈4.7. For large U, the optical spectral weight concentrates in incoherent peaks at ω ≈U. The inset of Fig. 3b shows the partial optical f-sums. As expected, both the contribution of the Drude peak and the total f-sum decrease for increasing U. Figure 4a shows the impact of the definition for σ(ω) on the results for U = 4.0. Our isotropic model yields by far the largest contributions 7 a) b) 0 0.5 1 1.5 0 1 2 3 σ(ω) |ω| U=4.0 isotropic disordered stacked 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 8 ∫0 ω dω’ σ(ω’) ω 0 0.1 0.2 0.3 0.4 0.5 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 ∫0 ∞ dω σ(ω) U -π/2 Ekin isotropic disordered stacked Figure 4. a) Optical conductivity σ(ω) for T = 0.05 and U = 4.0 for the new isotropic model with semi-elliptic DOS in comparison with disordered and stacked models consistent with the same DOS. b) Optical f-sum for T = 0.05. at small ω. While the stacked model leads to otherwise similar results, the low-frequency form of σ(ω) is qualitatively different in the disordered case (where a Drude peak is absent even for U →0). As seen in Fig. 4b, the f-sum is generally larger for the isotropic than for the stacked Bethe lattice, in particular in the metallic phase (while the disordered case is in-between). These differences can be attributed to the enhanced squared Fermi velocity in the isotropic model: The enhancement is largest near the Fermi surface, at ǫ = 0 by a factor of π/2 ≈1.57. A corresponding increase is expected of the Drude peak for small enough U and T when transport is dominated by states with ǫ ≈0. Since energy eigenstates spread out in momentum space at large U, the enhancement reduces (in general) to the integral R ∞ −∞dǫ ˜ρ(ǫ) = P∞ D=1 D t∗ D 2 which evaluates here to 1.05406. In all three cases,", + "and T when transport is dominated by states with ǫ ≈0. Since energy eigenstates spread out in momentum space at large U, the enhancement reduces (in general) to the integral R ∞ −∞dǫ ˜ρ(ǫ) = P∞ D=1 D t∗ D 2 which evaluates here to 1.05406. In all three cases, the proportionality (4) of the f-sum to the kinetic energy as characteristic for the hc lattice is clearly violated. This is true even for the stacked case (which is otherwise similar to the hc case): while the f-sum is here proportional to the contribution to the kinetic energy associ- ated with hopping in current direction [8], R ∞ 0 dω σxx(ω) = −σ0⟨ˆTx⟩/4, this contribution (which is negligible in the limit Z →∞) is not proportional to the total kinetic energy in this anisotropic case. A more relevant sum rule is derived from (12): R ∞ 0 dω σxx(ω) = t2a2σ0 4 ⟨−ǫ 4−ǫ2⟩. 4. Conclusion We have presented a new general method for constructing regular lat- tice models with hypercubic (hc) symmetry, i.e. isotropic optical transport properties, in large dimensions. Previously, calculations of the optical con- 8 ductivity σ(ω) of the Hubbard model in the limit d →∞had been restricted to the hypercubic lattice (using NCA [3] or QMC [10]) or have ignored the lattice dependence: Applying the hc formalism to the Bethe semi-elliptic DOS, Rozenberg et. al [11] overlooked violations of the hc f-sum rule. All previous approaches specific to the Bethe DOS were associated with anisotropic or incoherent transport; the most interesting of these [4, 6] have not yet been linked rigorously to microscopic models. Our method yields the first derivation for σ(ω) consistent with a semi- elliptic DOS that implies isotropic transport which is fully coherent in the noninteracting limit. This reinterpretation of the “Bethe lattice” (in the DMFT sense) as an isotropic, regular and clean lattice and the demon- stration that the associated transport properties are robust (with respect to finite dimensionality or hopping range) removes, finally, the pathologies previously associated with the DMFT treatment of transport in connection with non-Gaussian DOSs. At essentially no additional cost, the method can also be used for computing properties such as transverse conductivities and thermopower; these vanish, however, in the particle-hole symmetric case considered in this paper. Our numerical results have shown that the precise definition of σ(ω) does matter, in particular within the metallic phase where transport is potentially most coherent. We have also found a general DMFT expression for the f-sum rule as well as a form specific to our new approach. Acknowledgements We gratefully acknowledge useful discussions with J. Freericks, D. Logan, A. Millis, and D. Vollhardt. References 1. J. Hubbard, Proc. Roy. Soc. London A276, 238 (1963). M. C. Gutzwiller, Phys. Rev. Lett. 10, 59 (1963). J. Kanamori, Prog. Theor. Phys. 30, 275 (1963). 2. A. Khurana, Phys. Rev. Lett. 64, 1990 (1990). 3. T. Pruschke, D. L. Cox, and M. Jarrell, Phys. Rev. B 47, 3553 (1993). 4. W. Chung and J. K. Freericks, Phys. Rev. B 57, 11955 (1998). J. K. Freericks, private communication", + "J. Kanamori, Prog. Theor. Phys. 30, 275 (1963). 2. A. Khurana, Phys. Rev. Lett. 64, 1990 (1990). 3. T. Pruschke, D. L. Cox, and M. Jarrell, Phys. Rev. B 47, 3553 (1993). 4. W. Chung and J. K. Freericks, Phys. Rev. B 57, 11955 (1998). J. K. Freericks, private communication (2000, 2002). 5. N. Bl¨umer, Ph.D. Thesis, Universit¨at Augsburg, 2002. 6. A. Chattopadhyay, A. J. Millis, and S. Das Sarma, Phys. Rev. B 61, 10738 (2000). A. J. Millis, private communication (2002). 7. M. P. H. Stumpf, Ph.D. Thesis, University of Oxford (1999). 8. G. S. Uhrig and R. Vlaming, J. Phys. Cond. Matter 5, 2561 (1993). 9. V. Dobrosavljevi´c and G. Kotliar, Phys. Rev. Lett. 71, 3218 (1993). 10. M. Jarrell, J. K. Freericks, and T. Pruschke, Phys. Rev. B 51, 11704 (1995). 11. M. J. Rozenberg et. al, Phys. Rev. Lett. 75, 105 (1995).", + "arXiv:cond-mat/0008050v2 [cond-mat.str-el] 9 Aug 2000 Pseudogap and Kinetic Pairing Under Critical Differentiation of Electrons in Cuprate Superconductors Masatoshi Imada1) and Shigeki Onoda Institute for Solid State Physics, University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba, 277-8581, Japan 1) e-mail address, imada@issp.u-tokyo.ac.jp Abstract Superconducting mechanism of cuprates is discussed in the light of the proxim- ity of the Mott insulator. The proximity accompanied by suppression of coherence takes place in an inhomogeneous way in the momentum space in finite-dimensional systems. Studies on instabilities of metals consisted of such differentiated electrons in the momentum space are reviewed from a general point of view. A typical ex- ample of the differentiation is found in the flattening of the quasiparticle dispersion discovered around momenta (π, 0) and (0, π) on 2D square lattices. This flattening even controls the criticality of the metal-insulator transition. Such differentiation and suppressed coherence subsequently cause an instability to the superconduct- ing state in the second order of the strong coupling expansion. The d-wave pair- ing interaction is generated from such local but kinetic processes in the absence of disturbance from the coherent single-particle excitations. The superconducting mechanism emerges from a direct kinetic origin which is conceptually different from the pairing mechanism mediated by bosonic excitations as in magnetic, excitonic, and BCS mechanisms. Pseudogap phenomena widely observed in the underdoped cuprates are then naturally understood from the mode-mode coupling of d-wave su- perconducting (dSC) fluctuations repulsively coupled with antiferromagnetic (AFM) ones. When we assume the existence of a strong d-wave channel repulsively compet- ing with AFM fluctuations under the formation of flat and damped single-particle dispersion, we reproduce basic properties of the pseudogap seen in the magnetic resonance, neutron scattering, angle resolved photoemission and tunneling mea- surements in the cuprates. 1 Introduction Magnetism in strongly correlated electron systems has been a subject of extensive studies for a long time. In many examples, the energy gain from electron kinetic energy term is a crucial driving force for stabilizing magnetically symmetry broken states. Even in the Mott insulating state, this is equally true. Although the itinerancy of electrons is lost by the Coulomb interaction in the Mott insulator, electrons still find an optimized way to gain the energy from the kinetic energy term through a virtual process, where the strong coupling expansion becomes an adequate description under totally suppressed coherent 1 motion of electrons. Anderson’s mechanism of the superexchange interaction thus pro- posed [1] has long been one of the most fruitful concepts in the condensed matter physics. The double exchange mechanism proposed by de Gennes [2], Zenner [3] and Anderson and Hasegawa [4] is another example, where a ferromagnetic (FM) metal is stabilized by the kinetic energy gain under a strong Hund’s rule coupling between degenerate orbitals. Superconductivity, on the other hand, has been analyzed mainly from a different view since the dramatic success of BCS theory [5]. Various types of pairing interactions medi- ated by bosonic excitations have been considered to be responsible for the mechanisms. To say nothing of the electron-phonon interaction, roles of AFM and FM paramagnons and excitons have been", + "analyzed mainly from a different view since the dramatic success of BCS theory [5]. Various types of pairing interactions medi- ated by bosonic excitations have been considered to be responsible for the mechanisms. To say nothing of the electron-phonon interaction, roles of AFM and FM paramagnons and excitons have been studied in various contexts. In these proposals, the superconduct- ing ground state is the consequence of pair formation by the mediated attraction in the presence of coherent and metallic quasiparticles. In this report, we discuss in more general perspective the importance of the kinetic origin for various symmetry breakings. We particularly discuss the kinetic mechanism of the superconductivity under the suppressed coherence of quasiparticles near the Mott insulator. The mechanism of superconductivity we discuss in this paper is conceptually different from the conventional magnetic mechanism discussed in the spin fluctuation the- ories and in the t-J model, because the origin is not in the mediated bosons but in the direct kinetic process as in the superexchange for the mechanism of antiferromagnetism. Although the superexchange interaction is the only process for the energy lowering by the kinetic energy term in the Mott insulating phase, this constraint is lost in the pres- ence of doped carriers and various other processes may equally or even more importantly contribute to the kinetic energy gain. The existence of various other processes is in fact easily confirmed by the strong coupling expansion of the Hubbard-type models, where in the second order, namely in the order of t2/U, it not only generates the superexchange interaction but also other two-particle terms including the pair hopping processes. Here in the strong coupling expansion, the onsite Coulomb repulsion U is assumed to be larger than the kinetic transfer term t. These second-order terms usually have secondary impor- tance in the presence of the kinetic energy term in the first order of t in the normal metal. However, as similarly to the superexchange interaction in the Mott insulating phase, they become the primary origin of the kinetic energy gain even in metals when the single- particle processes are suppressed by some reason but the amplitude of the second-order terms are retained. In the study on the strong-coupling expansion of the d-p model, it was shown that the d-wave pair-hopping process appears in the third order, although the superexchange interaction appears only in the fourth order [6]. In the next section we first discuss the origin of the unusual suppression of coherent single-particle motion in the proximity of the Mott insulator before considering the mechanism of superconductivity. High-temperature cuprate superconductors show a variety of unusual properties in the 2 normal state [7]. Among others, we concentrate on two remarkable properties widely ob- served in the cuprates for the purpose of examining the relevance of the kinetic mechanism as the driving mechanism of the high-Tc superconductivity. One of the remarkable prop- erties in the cuprates is the flat dispersion in single-particle excitations. Angle resolved photoemission spectra (ARPES) in Y and Bi based high-Tc cuprates show an unusual dispersion which is far from weak correlation picture", + "the kinetic mechanism as the driving mechanism of the high-Tc superconductivity. One of the remarkable prop- erties in the cuprates is the flat dispersion in single-particle excitations. Angle resolved photoemission spectra (ARPES) in Y and Bi based high-Tc cuprates show an unusual dispersion which is far from weak correlation picture [8]. The dispersion around (π, 0) and (0, π) is extremely flat beyond the expectations from usual van Hove singularities. The flat dispersion also shows rather strong damping. The other remarkable property we discuss is the pseudogap phenomenon observed in the underdoped region [7, 9]. It is observed both in spin and charge excitations in which gap structure emerges from a temperature TPG well above the superconducting transition point Tc. The gap structure is observed in various different probes such as NMR relaxation time, the Knight shift, neutron scattering, tunneling, ARPES, specific heat, optical conductivity, and DC resistivity. The ARPES [10, 11] data have revealed that the pseudogap starts growing first in the region around (π, 0) and (0, π) from T = TPG much higher than Tc. Therefore, the pseudogap appears from the momentum region of the flattened dispersion and it is likely that the mechanism of the pseudogap formation is deeply influenced from the underlying flatness. The superconducting state itself also shows a dominant gap structure in this flat spots, (π, 0) and (0, π), due to the dx2−y2 symmetry. In fact, the pseudogap structure above Tc appears continuously to merge into the dx2−y2 gap below Tc. To understand the superconducting mechanism and the origin of the high transition temperatures, a detailed understanding of the physics taking place in the flat dispersion region is required. The emergence of the flat dispersion around (π, 0) and (0, π) has also been reported in numerical simulation results rather universally in models for strongly correlated electrons such as the Hubbard and t-J models on square lattices [12, 13, 14]. As we see in the next section, this criticality is interpreted from a strong proximity of the Mott insulator where strong electron correlation generates suppressed dynamics and coherence. 2 Kinetic Pairing Derived from Electron Differenti- ation In a simple picture, the correlation effects emerge as the isotropic mass renormalization, where the Coulomb repulsion from other electrons makes the effective mass heavier. This effect was first demonstrated by Brinkman and Rice [19] in the Gutzwiller approximation and refined in the dynamical mean field theory [20]. In the numerical results on a square lattice, as discussed above, the correlation ef- fects appear in more subtle way where the electrons at different momenta show different 3 renormalizations. When the Mott insulator is approached and the doping concentration becomes small, the mass renormalization generally becomes stronger. However, once the renormalization effect gets relatively stronger in a part of the Fermi surface, it is further enhanced at that part in a selfconsistent fashion because the slower electrons become more and more sensitive to the correlation effect. This generates critical differentiation of the carriers depending on the portion of the Fermi surface. On square lattices, the stronger renormalization happens around", + "the Fermi surface, it is further enhanced at that part in a selfconsistent fashion because the slower electrons become more and more sensitive to the correlation effect. This generates critical differentiation of the carriers depending on the portion of the Fermi surface. On square lattices, the stronger renormalization happens around (π, 0) and (0, π). A part of this anisotropic correlation effect concentrating near (π, 0) and (0, π) is intuitively understood from the carrier motion under the background of AFM correlations. As we see a real space picture in Fig.1a, the carrier motion in the diagonal directions does not disturb the correlations due to the parallel spin alignment, while the motion in horizontal and vertical directions strongly disturbs the AFM backgrounds as expressed as the wavy bonds and the motion itself is also disturbed as a feedback. Such strong coupling of charge dynamics to spin correlations causes flattening and damping of electrons around (π, 0) and (0, π), but not around the diagonal direction (±π/2, ±π/2). The anisotropic renormalization effect eventually may generate a singularly flat dispersion on particular region of the Fermi surface, which accepts more and more doped holes in that region due to the enhanced density of states. The transition to the Mott insulator is then governed by that flattened part, since the carriers reside predominantly in the flat region. The criticality of the metal-insulator transition on the square lattice is thus determined from the doped carriers around the flat spots, (π, 0) and (0, π). The hyperscaling relation becomes naturally satisfied because singular points on the momentum space govern the transition. In fact, the hyperscal- ing relations are numerically supported in various quantities and shows agreements with experimental indications. For example, the electronic compressibility critically diverges as κ ∝1/δ with decreasing doping concentration δ while the Drude weight is unusually suppressed as D ∝δ2 [15, 16]. The coherence temperature (the effective Fermi tempera- ture) is also scaled as TF ∝δ2 and indicates unusual suppression. In more comprehensive understanding, all the numerical data are consistent with the hyperscaling relations with a large dynamical exponent z = 4 for the metal-insulator transition [7, 18]. Such large exponent opposed to the usual value z = 2 for the transition to the band insulator is derived from the slower electron dynamics even at T = 0 generated by the flat dispersion. We, however, should keep in mind that the relaxation time of quasiparticles and the damping constant of magnetic excitations do not have criticality at the transition point to the Mott insulator. A general remark is that the relaxation time is critical only in the case of the Anderson localization transition and not in the case of the transition to the Mott insulator. The DC transport properties and magnetic relaxation phenomena are contaminated by such noncritical relaxation times τ and are influenced by the carriers 4 in the other portion than the flat part because the flat part has stronger damping and less contributes to the DC properties. Large anisotropy of τ masks the real criticality and makes it difficult to", + "are contaminated by such noncritical relaxation times τ and are influenced by the carriers 4 in the other portion than the flat part because the flat part has stronger damping and less contributes to the DC properties. Large anisotropy of τ masks the real criticality and makes it difficult to see the real critical exponents in the τ-dependent properties. Relevant quantities to easily estimate the criticality is the τ independent quantities such as the Drude weight and the compressibility. Near the metal-insulator transition, the critical electron differentiation and selective renormalization may lead to experimental observations as if internal degrees of freedom of the carriers such as spin and charge were separated because each degrees of freedom can predominantly be conveyed by carriers in different part of the Fermi surface. An- other possible effect of the electron differentiation is the appearance of several different relaxation times which are all originally given by a single quasiparticle relaxation in the isotropic Fermi liquids, but now depend on momenta of the quasiparticles. Another aspect one might ask in connection to the relevance of the flat part to the metal-insulator transition is the observed level difference between (π, 0) and (±π/2, ±π/2) in the undoped and underdoped cuprates [17]. The level at (π, 0) is substantially far from the Fermi level, and at a first glance, it does not have a chance to contribute to the metal- insulator transition. However, it has been clarified [6] that this level difference is likely to be absent before the d-wave interaction channel starts growing where the flat part indeed governs the criticality, while it is developed by renormalization of the single-particle level accompanied by the d-wave interaction in the lower energy scale. It is remarkable that this renormalization exists even in the insulator. If the mass renormalization would happen in an isotrpic way as in the picture of Brinkman and Rice, the renormalization can become stronger without disturbance when the insulator is approached. However, if the singularly renormalized flat dispersion emerges critically only in a part near the Fermi surface but the whole band width is ratained, that flattened part has stronger instability due to the coupling to larger energy scale retained in other part of the momentum space. The instability can be mediated by local and incoherent carrier motion generated from two-particle processes derived in the strong coupling expansion [6]. The local two-particle motion is given in the order of t2/U with the bare t while the single-particle term is renormalized to t∗which can be smaller than t2/U at the flattened portion. The instability of the flat dispersion was studied by taking account such local and incoherent terms in the Hubbard and t-J mod- els [21, 22, 16]. The inclusion of the two-particle terms drives the instability of the flat part to the superconducting pairing and the formation of the d-wave gap structure. In fact, even at half filling, the two-particle process stabilizes the d-wave superconducting state and reproduces the basic feature of the pseudogap formation [21, 22] observed in the BEDT-TTF compounds [23]. The paired bound particles formed", + "flat part to the superconducting pairing and the formation of the d-wave gap structure. In fact, even at half filling, the two-particle process stabilizes the d-wave superconducting state and reproduces the basic feature of the pseudogap formation [21, 22] observed in the BEDT-TTF compounds [23]. The paired bound particles formed from two quasiparticles at the flat spots have dif- 5 ferent dynamics from the original quasiparticle. In fact, when the paired singlet becomes the dominant carrier, the criticality changes from z = 4 to z = 2, resulting in the recovery of coherence and kinetic energy gain [16]. It generates a strong pairing interaction from the kinetic origin. This pairing mechanism is a consequence of suppressed single-particle coherence and electron differentiation due to strong correlations. The instability of the flat dispersion coexisting with relatively large incoherent process was further studied [6, 24, 25, 26]. It has turned out that promotion of the above scaling behavior and the flat dispersion offers a way to control potential instabilities. Even when a flat band dispersion is designed near the Fermi level by controlling lattice geometry and parameters, it enlarges the critical region under the suppression of single-particle coherence in the proximity of the Mott insulator mentioned above. In designed lattices and lattices with tuned lattice parameters, it was reported that the superconducting instability and the formation of the spin gap have been dramatically enhanced [24]. 3 Pseudogap Phenomena in Cuprates as Supercon- ducting Fluctuations As is mentioned in §1, the pseudogap in the high-Tc cuprates starts growing from the region of the flat dispersion. When the single-particle coherence is suppressed, the system is subject to two particle instabilities. As clarified in §2, the superconducting instability in fact grows. However, the AFM and charge order correlations are in principle also expected to grow from other two-particle (particle-hole) processes and may compete each other. In particular, the AFM long-range order is realized in the Mott insulator and its short-range correlation is well retained in the underdoped region. Therefore, to understand how the superconducting phase appears in the underdoped region, at least competition of dSC and AFM correlations has to be treated with underlying suppressed coherence in the region of (π, 0) and (0, π). The authors have developed a framework to treat the competition by employing the mode-mode coupling theory of dSC and AFM fluctuations where these two fluctuations are treated on an equal footing [27, 28]. It should be noted that the strong dSC pairing interaction is resulted from a highly correlated effect with electron differentiation while the critical differentiation has not been successfully reproduced from the diagrammatic approach so far. Then, within the framework of the mode-mode coupling theory, at the starting point, we have assumed the existence of correlation effects leading to the flattened dispersion and the d-wave pair hopping process. The AFM and dSC fluctuations are predominantly generated by the contributions from the quasiparticle excitations in the flattened regions (π, 0) and (0, π). These fluctuations are treated in a set of selfconsistent equations with mode couplings of dSC and AFM. From the", + "dispersion and the d-wave pair hopping process. The AFM and dSC fluctuations are predominantly generated by the contributions from the quasiparticle excitations in the flattened regions (π, 0) and (0, π). These fluctuations are treated in a set of selfconsistent equations with mode couplings of dSC and AFM. From the selfconsistent solution, the pseudogap formation is well repro- 6 duced in a region of the parameter space. The pseudogap emerges when the mode coupling between dSC and AFM is repulsive with a severe competition and dSC eventually domi- nates at low temperatures. Such competition suppresses Tc, while above Tc it produces a region where pairing fluctuations are large. This region at TP G > T > Tc shows suppres- sion of 1/T1T and the pseudogap formation around (π, 0) and (0, π) in A(k, ω). These reproduce the basic feature of the pseudogap phenomena experimentally observed in the underdoped cuprates. The pseudogap formation is identified as coming from the super- conducting fluctuations. The momentum dependence shows that the pseudogap formation starts around (π, 0) from higher temperatures and the formation temperature becomes lower with increasing distance from (π, 0) . All of the above reproduce the experimental observations. We, however, note a richer structure of the gap formation observed in the transversal NMR relaxation time T2G and the neutron resonance peak. One puzzling experimental observation is that the pseudogap structure appears in 1/T1T [9, 29, 30, 31, 32], while in many cases 1/T2G, which measures Reχ(Q, ω = 0) at Q = (π, π), continuously increases with the decrease in temperature with no indication of the pseudogap. In addition, the so called resonance peak appears in the neutron scattering experiments [33]. A resonance peak sharply grows at a finite frequency below Tc with some indications even at Tc < T < TPG. This peak frequency ω∗decreases with lowering doping concentration implying a direct and continuous evolution into the AFM Bragg peak in the undoped compounds. The neutron and T2G data support the idea that the AFM fluctuations are suppressed around ω = 0 but transferred to a nonzero frequency below TPG. To understand these features, a detailed consideration on damping of the magnetic excitations is required. With the increase in the pairing correlation length ξd, the pseu- dogap in A(k, ω) is developed. Since the damping is mainly from the overdamped Stoner excitations, the gap formation in A(k, ω) contributes not only to suppress growth of AFM correlation length ξσ but also to reduce the magnetic damping because, inside the domain of the d-wave order, the AFM excitations are less scattered due to the absence of low- energy quasiparticle around (π, 0). If the quasiparticle damping is originally large around (π, 0), the damping γ can be reduced dramatically upon the pseudogap formation. Under this circumstance, our calculated result reproduces the resonance peak and the increase in 1/T2G with lowering temperature at T > Tc in agreement with the experimental obser- vations in YBa2Cu3O6.63, YBa2Cu4O8 and some other underdoped compounds [27, 28]. A subtlety arises when the damping around (π/2, π/2) starts", + "formation. Under this circumstance, our calculated result reproduces the resonance peak and the increase in 1/T2G with lowering temperature at T > Tc in agreement with the experimental obser- vations in YBa2Cu3O6.63, YBa2Cu4O8 and some other underdoped compounds [27, 28]. A subtlety arises when the damping around (π/2, π/2) starts contributing. This is particularly true under the pseudogap formation. If contributions from the (π/2, π/2) region would be absent, the damping of the magnetic excitation would be strongly reduced when the pseudogap is formed around (π, 0) as we mentioned above. However, under the pseudogap formation, the damping can be determined by the Stoner continuum generated 7 from the (π/2, π/2) region and can remain overdamped. This process is in fact important if the quasiparticle damping around the (π/2, π/2) region is large as in the case of La 214 compounds [34]. The formation of the pseudogap itself is a rather universal consequence of the strong coupling superconductors. However, the actual behavior may depend on this damping. If the damping generated by the (π/2, π/2) region is large, it sensitively destroy the resonance peak structure observed in the neutron scattering experimental results. 4 Conclusion and Discussion Electron critical differentiation is a typical property of the proximity of the Mott insulator. The flattening of the quasiparticle dispersion appears around momenta (π, 0) and (0, π) on square lattices and determines the criticality of the metal-insulator transition with the suppressed coherence in that momentum region of quasiparticles. Such coherence suppression subsequently causes an instability to the superconducting state when a proper incoherent kinetic process is retained. The d-wave superconducting state is stabilized from such retained microscopic process derived from the strong correlation expansion. The origin of the superconductivity is ascribed to the kinetic energy gain. By assuming the d-wave channel and the presence of strongly renormalized flat quasi- particle dispersion around the (π, 0) region, we have constructed the mode-mode coupling theory for the AFM and dSC fluctuations. The pseudogap in the high-Tc cuprates is re- produced as the region with enhanced dSC correlations and is consistently explained from precursor effects for the superconductivity. The existence of the flat region plays a role to suppress the effective Fermi temperature EF. This suppressed EF and relatively large local pair hopping process both drive the system to the strong coupling superconductor thereby leading to the pseudogap formation. The pseudogap formation is also enhanced by the AFM fluctuations repulsively coupled with dSC fluctuations. Several similar attempts have also been made to reproduce the pseudogap phenomena observed in the cuprates. A common conclusion inferred from these calculations including those by the present authors is that the pseudogap is reproduced when the d-wave channel is explicitly assumed [27, 28, 35] while crucial features such as the pseudogap formation around (π, 0) cannot be well reproduced if one tries to derive the superconducting channel itself from the AFM spin fluctuations [36]. This difficulty is summarized: if one desires to stabilize a strongly fluctuating pseudogap region well above Tc generated originally by the magnetic interaction, one has to treat the strong-coupling", + "(π, 0) cannot be well reproduced if one tries to derive the superconducting channel itself from the AFM spin fluctuations [36]. This difficulty is summarized: if one desires to stabilize a strongly fluctuating pseudogap region well above Tc generated originally by the magnetic interaction, one has to treat the strong-coupling superconductivity and hence even more strong-coupling magnetic interaction. In such a circumstance, it is dificult to escape from the magnetic instability before the pseudogap formation. The difficulty is naturally interpreted from our analysis: The kinetic d-wave channel we derived in the strong coupling expansion is not ascribed to the magnetic origin and is not contained 8 neither in the spin-fluctuation mechanism nor in the t-J model. This superconducting channel has a comparable amplitude to the magnetic one, namely t2/U. Furthermore the channel is enhanced by the flattened dispersion and the electron critical differentiation. Such effects are far beyond the one-loop level of the weak coupling approach. The change in the quasiparticle dispersion in the portion of the Fermi surface is not well reproduced for the moment and the momentum dependent selfenergy appears to be significantly underestimated in the existing diagrammatic evaluations. The strong coupling Hamiltonian was considered before [37] and also in terms of the pairing arising from the attractive superexchange interaction [38]. The superconductivity from direct kinetic origin was also discussed before in a specific model [39] as well as in the interlayer tunneling mechanism [40], where interlayer kinetic energy gain is required to stabilize the superconductivity. If we identify the intralayer charge incoherence observed in the normal state as the crossover phenomena above the unusually suppressed Fermi temperature in the proximity of the Mott insulator, and notice that no quantum coherence is reached even within the layer, the kinetic origin of the superconductivity is found solely in a two-dimensional plane. Under the suppression of coherence in the proximity of the Mott insulator, the second- order process in the strong-coupling expansion may also drive other type of ordered state in addition to the superconductivity. In particular, because the charge compressibility is enhanced due to the flattened dispersion, the charge fluctuation may be strongly en- hancced and the charge ordering and stripes can be triggered both by the kinetic energy gain and the intersite Coulomb interaction. When the competition between the supercon- ductivity and the charge ordering becomes serious, presumable repulsive mode coupling of them may become another origin of the pseudogap formation, although such evidence is not clear in the present experimental results of the cuprates. The presend theory of the kinetic superconductor predicts a specific form of the kinetic energy gain to be seen in the optical conductivity [41] and the single particle spectra [42]. Qualitatively, it is expected that the in-plane kinetic energy starts gained even in the pseudogap region. 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In (b), frustrations are generated after the hole motion (denoted by circles) as shown by wavy bonds. These differences induce the differentiation between electrons around (π, 0) and (π/2, π/2). 12", + "arXiv:physics/0503206v3 [physics.gen-ph] 26 Sep 2006 Neutrinos in the Electron E.L. Koschmieder Center for Statistical Mechanics The University of Texas at Austin, Austin TX 78712, USA e-mail: koschmieder@mail.utexas.edu We will show that one half of the rest mass of the electron is equal to the sum of the rest masses of electron neutrinos and that the other half of the rest mass of the electron is given by the energy in the sum of electric oscillations. With this composition we can explain the rest mass, the electric charge, the spin and the magnetic moment of the electron. Introduction After J.J. Thomson [1] discovered the small corpuscle which soon became known as the electron an enormous amount of theoretical work has been done to explain the existence of the electron. Some of the most distinguished physicists have participated in this effort. Lorentz [2], Poincar´e [3], Ehrenfest [4], Einstein [5], Pauli [6], and others showed that it is fairly certain that the electron cannot be explained as a purely electromagnetic particle. In particular it was not clear how the electrical charge could be held together in its small volume because the internal parts of the charge repel each other. Poincar´e [7] did not leave it at showing that such an electron could not be stable, but suggested a solution for the problem by introducing what has become known as the Poincar´e stresses whose origin however remained unexplained. These studies were concerned with the static properties of the electron, its mass m(e±) and its electric charge e. In order to explain the electron with its existing mass and charge it appears to be necessary to add to Maxwell’s equations a non-electromagnetic mass and a non-electromagnetic force which could hold the electric charge together. We shall see what this mass and force is. The discovery of the spin of the electron by Uhlenbeck and Goudsmit [8] increased the difficulties of the problem in so far as it now had also to be explained how the angular momentum ¯h/2 and the magnetic moment µe come about. The spin of a point-like electron seemed to be explained by 1 Dirac’s [9] equation, however it turned out later [10] that Dirac type equa- tions can be constructed for any value of the spin. Afterwards Schr¨odinger [11] tried to explain the spin and the magnetic moment of the electron with the so-called Zitterbewegung. Later on many other models of the electron were proposed. On p.74 of his book “The Enigmatic Electron” Mac Gregor [12] lists more than thirty such models. At the end none of these models has been completely successful because the problem developed a seemingly insur- mountable difficulty when it was shown through electron-electron scattering experiments that the radius of the electron must be smaller than 10−16 cm, in other words that the electron appears to be a point particle, at least by three orders of magnitude smaller than the classical electron radius re = e2/mc2 = 2.8179·10−13 cm. This, of course, makes it very difficult to explain how a particle can have a finite angular momentum when the", + "other words that the electron appears to be a point particle, at least by three orders of magnitude smaller than the classical electron radius re = e2/mc2 = 2.8179·10−13 cm. This, of course, makes it very difficult to explain how a particle can have a finite angular momentum when the radius goes to zero, and how an electric charge can be confined in an infinitesimally small volume. If the elementary electrical charge were contained in a volume with a radius of O(10−16) cm the Coulomb self-energy would be orders of magnitude larger than the rest mass of the electron, which is not realistic. The choice is be- tween a massless point charge and a finite size particle with a non-interacting mass to which an elementary electrical charge is attached. We propose in the following that the non-electromagnetic mass which seems to be necessary in order to explain the mass of the electron consists of neutrinos. This is actually a necessary consequence of our standing wave model [13] of the masses of the mesons and baryons. And we propose that the non-electromagnetic force required to hold the electric charge and the neutrinos in the electron together is the weak nuclear force which, as we have suggested in [13], holds together the masses of the mesons and baryons and also the mass of the muons. Since the range of the weak nuclear force is on the order of 10−16 cm the neutrinos can only be arranged in a lattice with the weak force extending from each lattice point only to the nearest neighbors. The size of the neutrino lattice in the electron does not at all contradict the results of the scattering experiments, just as the explanation of the mass of the muons with the standing wave model does not contradict the apparent point particle characteristics of the muon, because neutrinos are in a very good approximation non-interacting and therefore are not noticed in scattering experiments with electrons. 2 1 The mass and charge of the electron The rest mass of the electron is m(e±) = 0.510 998 92 ± 4·10−8 MeV/c2 and the electrostatic charge of the electron is e = 4.803 204 41·10−10 esu, as stated in the Review of Particle Physics [14]. Both are known with great accuracy. The objective of a theory of the electron must be the explanation of both values. We will first explain the rest mass of the electron making use of what we have learned from the standing wave model, in particular of what we have learned about the explanation of the mass of the µ± mesons in [13]. The muons are leptons, just as the electrons, that means that they interact with other particles exclusively through the electric force. The muons have a mass which is 206.768 times larger than the mass of the electron, but they have the same elementary electric charge as the electron or positron and the same spin. Scattering experiments tell that the µ± mesons are point particles with a size < 10−16 cm, just as the electron. In other words,", + "is 206.768 times larger than the mass of the electron, but they have the same elementary electric charge as the electron or positron and the same spin. Scattering experiments tell that the µ± mesons are point particles with a size < 10−16 cm, just as the electron. In other words, the muons have the same characteristics as the electrons and positrons but for a mass which is about 200 times larger. Consequently the muon is often referred to as a “heavy” electron. If a non-electromagnetic mass is required to explain the mass of the electron then a non-electromagnetic mass 200 times as large as in the electron is required to explain the mass of the muons. These non-electromagnetic masses must be non-interacting, otherwise scattering experiments could not find the size of either the electron or the muon at 10−16 cm. We have already explained the mass of the muons with the standing wave model [13]. According to this model the muons consist of an elementary electric charge and a lattice of neutrinos which, as we know, do not interact with charge or mass. Neutrinos are the only non-interacting matter we know of. In the muon lattice are, according to [13], (N - 1)/4 = N′/4 muon neutrinos νµ (respectively anti-muon neutrinos ¯νµ), N′/4 electron neutrinos νe and the same number of anti-electron neutrinos ¯νe, one elementary electric charge and the energy of the lattice oscillations. The letter N stands for the number of all neutrinos and antineutrinos in the cubic lattice of the π± mesons [13, Eq.(15)] N = 2.854 · 109 . (1) It is, according to [13], a necessary consequence of the decay of the µ−muon µ−→e−+ ¯νe + νµ that there must be N′/4 electron neutrinos νe in the emitted electron, where N′ = N - 1 ∼= N [13]. For the mass of the electron neutrinos and anti-electron neutrinos we found in Eq.(34) of [13] that 3 m(νe) = m(¯νe) = 0.365 milli eV/c2 . (2) The sum of the energies in the rest masses of the N′/4 neutrinos or antineu- trinos in the lattice of the electron or positron is then X m(νe)c2 = N′/4 · m(νe)c2 = 0.260 43 MeV = 0.5096 m(e±)c2 . (3) To put this in other words, one half of the rest mass of the electron comes from the rest masses of electron neutrinos. The other half of the rest mass of the electron must originate from the energy in the electric charge carried by the electron. From pair production γ + M →e−+ e+ + M, (M being any nucleus), and from conservation of neutrino numbers follows necessarily that there must also be a neutrino lattice composed of N′/4 anti-electron neutrinos, which make up the lattice of the positrons, which lattice has, because of Eq.(2), the same rest mass as the neutrino lattice of the electron, as it must be for the antiparticle of the electron. Fourier analysis dictates that a continuum of high frequencies must be in the electrons or positrons created by pair production in a timespan of 10−23", + "has, because of Eq.(2), the same rest mass as the neutrino lattice of the electron, as it must be for the antiparticle of the electron. Fourier analysis dictates that a continuum of high frequencies must be in the electrons or positrons created by pair production in a timespan of 10−23 seconds. We will now determine the energy Eν(e±) contained in the oscillations in the interior of the electron. Since we want to explain the rest mass of the electron we can only consider the frequencies of non-progressive waves, either standing waves or circular waves. The sum of the energies of the lattice oscillations is, in the case of the π± mesons, given by Eν(π±) = hν0N 2π(ehν/kT −1) π Z −π φ dφ . (4) This is Eq.(14) combined with Eq.(16) in [13] where they were used to de- termine the oscillation energy in the π0 and π± mesons. This equation was introduced by Born and v. Karman [15] in order to explain the internal en- ergy of cubic crystals. In Eq.(4) h is Planck’s constant, ν0 = c/2πa is the reference frequency with the lattice constant a = 10−16 cm, N is the number of all oscillations, φ = 2πa/λ and T is the temperature in the lattice, for which we found in [13] the value T = 2.38 · 1014 K. If we apply Eq.(4) to the oscillations in the electron which has N′/4 electron neutrinos νe we arrive at Eν(e±) = 1/4·Eν(π±), which is mistaken because Eν(π±) ≈m(π±)c2/2 and 4 m(π±) ≈273 m(e±). Eq.(4) must be modified in order to be suitable for the oscillations in the electron. It turns out that we must use Eν(e±) = hν0N · αf 2π(ehν/kT −1) π Z −π φ dφ , (5) where αf is the fine structure constant. The appearance of αf in Eq.(5) indicates that the nature of the oscillations in the electron is different from the oscillations in the π0 or π± lattices. With αf = e2/¯hc and ν0 = c/2πa we have hν0αf = e2/a (6) that means that the oscillations in the electron are electric oscillations. There must be N′/2 oscillations of the elements of the electric charge in e±, because we deal with non-progressive waves, the superposition of two waves. As we will see later the spin requires that the oscillations are circular. That means that 2×N′/4 ∼= N/2 oscillations are in Eq.(5). From Eqs.(4,5) then follows that Eν(e±) = αf/2 · Eν(π±) . (7) Eν(π±) is the oscillation energy in the π± mesons which can be calculated with Eq.(4). According to Eq.(27) of [13] it is Eν(π±) = 67.82 MeV = 0.486 m(π±)c2 ≈m(π±)c2/2 . (8) With Eν(π±) ≈m(π±)c2/2 = 139.57/2 MeV and αf = 1/137.036 follows from Eq.(7) that Eν(e±) = αf 2 · m(π±)c2 2 = 0.254 62 MeV = 0.99657 m(e±)c2/2 . (9) We have determined the value of the oscillation energy in e± from the product of the very accurately known fine structure constant and the very accurately known rest mass of the π± mesons. One half of the energy in", + "m(π±)c2 2 = 0.254 62 MeV = 0.99657 m(e±)c2/2 . (9) We have determined the value of the oscillation energy in e± from the product of the very accurately known fine structure constant and the very accurately known rest mass of the π± mesons. One half of the energy in the rest mass of the electron comes from the electric oscillations in the electron. The other half of the energy in the rest mass of the electron is in the rest masses of the neutrinos in the electron. We can confirm Eq.(9) using Eq.(5) or Eq.(13) with N/2 = 1.427·109, e = 4.803·10−10 esu, a = 1·10−16 cm, f(T) = 1/1.305·1013, and with the integral 5 being π2 we obtain Eν(e±) = 0.968 m(e±)c2/2. This calculation involves more parameters than Eq.(9) and is consequently less accurate than Eq.(9). In a good approximation the oscillation energy of e± in Eq.(9) is equal to the sum of the energies in the rest masses of the electron neutrinos in the e± lattice in Eq.(3). Since m(e±)c2 = Eν(e±) + X m(νe)c2 = Eν(e±) + N′/4 · m(νe)c2 , (10) it follows from Eqs.(3) and (9) that m(e±)c2(theor) = 0.5151 MeV = 1.0079 m(e±)c2(exp) . (11) The measured rest mass of the electron or positron agrees within the accuracy of the parameters N and m(νe) with the theoretically predicted rest masses. From Eq.(7) follows with Eν(π±) ∼= m(π±)c2/2 that 2Eν(e±) ∼= m(e±)c2 = αfEν(π±) = αfm(π±)c2/2 , or that m(e±) · 2/αf = 274.072 m(e±) ∼= m(π±) , (12) whereas the actual ratio of the mass of the π± mesons to the mass of the electron is m(π±)/m(e±) = 273.132 or 0.9965·2/αf. We have here recovered the ratio m(π±)/m(e±) which we found with the standing wave model of the π± mesons, Eq.(65) of [13]. This seems to be a necessary condition for the validity of our model of the electron. We have thus shown that the rest mass of the electron can be explained by the sum of the rest masses of the electron neutrinos in a cubic lattice with N′/4 electron neutrinos νe and the mass in the sum of the energy of N/2 electric oscillations in the lattice, Eq.(9). The one oscillation added to the 2×N′/4 oscillations is the oscillation at the center of the lattice, Fig.(1). From this model follows, since it deals with a cubic neutrino lattice, that the electron is not a point particle, which is unlikely to begin with, because at a true point the self-energy would be infinite. However, since neutrinos are non-interacting their presence will not be detected in electron-electron scattering experiments. The rest mass of the muon has been explained similarly with an oscillating lattice of muon and electron neutrinos [13]. We found that m(µ±)/m(e±) is ∼= 3/2αf = 205.55, nearly equal to the actual mass ratio 206.768, in agreement 6 Fig.1. Horizontal or vertical section through the central part of the electron lattice. with what Nambu [16] found empirically. The heavy weight of the muon is primarily a consequence of the heavy weight of the N′/4", + "3/2αf = 205.55, nearly equal to the actual mass ratio 206.768, in agreement 6 Fig.1. Horizontal or vertical section through the central part of the electron lattice. with what Nambu [16] found empirically. The heavy weight of the muon is primarily a consequence of the heavy weight of the N′/4 muon neutrinos in the muon lattice. The mass of the muon neutrino is related to the mass of the electron neutrino through m(νe) = αfm(νµ), Eq.(39) of [13]. In order to confirm the validity of our preceding explanation of the mass of the electron we must show that the sum of the charges of the electric oscillations in the interior of the electron is equal to the elementary electric charge of the electron. We recall that Fourier analysis requires that, after pair production, there must be a continuum of frequencies in the electron and positron. With hν0αf = e2/a from Eq.(6) follows from Eq.(5) that the oscillation energy in e± is the sum of 2×(N′/4 + 1) ∼= N/2 electric oscillations Eν(e±) = N 2 · e2 a · f(T) 2π π Z −π φ dφ , (13) with f(T) = 1/(ehν/kT−1) = 1/1.305·1013 from p.17 in [13]. Inserting the values for N, f(T) and a we find that Eν(e±) = 0.968 m(e±)c2/2. The dis- crepancy between m(e±)c2/2 and Eν(e±) so calculated must originate from the uncertainty of the parameters N, f(T) and a in Eq.(13). We note that it follows from the factor e2/a in Eq.(13) that the oscillation energy is the same for electrons and positrons, as it must be. 7 We replace the integral divided by 2π in Eq.(13), which has the value π/2, by the sum Σ φk∆φ, where k is an integer number with the maximal value km = (N/4)1/3. φk is equal to kπ/km and we have Σ φk ∆φ = km X k=1 kπ km · 1 km = km(km + 1)π 2 k2 m ∼= π 2 , as it must be. The energy in the individual electric oscillation with index k is then ∆Eν(k) = φk ∆φ = kπ/k2 m . (14) Suppose that the energy of the electric oscillations is correctly described by the self-energy of an electrical charge U = 1/2 · e2/r . (15) The self-energy of the elementary electrical charge is normally used to deter- mine the mass of the electron from its charge, here we use Eq.(15) the other way around, we determine the charge from the energy in the oscillations. The charge of the electron is contained in the electric oscillations. That means that the electric charge is not concentrated in a point but is dis- tributed over N/4 = O(109) charge elements Qk. The charge elements are distributed in a cubic lattice and the resulting electric field is cubic, not spher- ical. For distances large as compared to the sidelength of the cube, (which is O(10−13) cm), say at the first Bohr radius which is on the order of 10−8 cm, the deviation of the cubic field from the spherical field will be reduced by about", + "field is cubic, not spher- ical. For distances large as compared to the sidelength of the cube, (which is O(10−13) cm), say at the first Bohr radius which is on the order of 10−8 cm, the deviation of the cubic field from the spherical field will be reduced by about 10−10. The charge in all electric oscillations is Q = X k Qk . (16) Setting the radius r in the formula for the self-energy equal to 2 a we find, with Eqs.(13,14,15), that the charge in the individual electric oscillations is Qk = ± q 2π N e2f(T)/k2m · √ k . (17) and with km = 1/2 (N/4)1/3 = 447 and km X k=1 √ k = 6310.8 8 follows, after we have doubled the sum over √ k, because for each index k there is a second oscillation on the negative axis of φ, that Q = Σ Qk = ± 5.027 · 10−10 esu , (18) whereas the elementary electrical charge is e = ± 4.803 · 10−10 esu. That means that our theoretical charge of the electron is 1.047 times the elemen- tary electrical charge. Within the uncertainty of the parameters the theoret- ical charge of the electron agrees with the experimental charge e. We have confirmed that it follows from our explanation of the mass of the electron that the electron has, within a 5% error, the correct electrical charge. Each element of the charge distribution is surrounded in the horizontal plane by four electron neutrinos as in Fig.(1), and in vertical direction by an electron neutrino above and also below the element. The electron neutrinos hold the charge elements in place. We must assume that the charge elements are bound to the neutrinos by the weak nuclear force. The weak nuclear force plays here a role similar to its role in holding, for example, the π± or µ± lattice together. It is not possibe, in the absence of a definitive explanation of the neutrinos, to give a theoretical explanation for the electro-weak interaction between the electric oscillations and the neutrinos. However, the presence of the range a of the weak nuclear force in e2/a is a sign that the weak force is involved in the electric oscillations. The attraction of the charge elements by the neutrinos overcomes the Coulomb repulsion of the charge elements. The weak nuclear force is the missing non-electromagnetic force or the Poincar´e stress which holds the elementary electric charge together. The same considerations apply for the positive electric charge of the positron, only that then the electric oscillations are all of the positive sign and that they are bound to anti-electron neutrinos. Finally we learn that Eq.(13) precludes the possibility that the charge of the electron sits only on its surface. The number N in Eq.(13) would then be on the order of 106, whereas N must be on the order of 109 so that Eν(e±) can be m(e±)c2/2 as is necessary. In other words, the charge of the electron must be distributed throughout the interior of the electron, as we", + "The number N in Eq.(13) would then be on the order of 106, whereas N must be on the order of 109 so that Eν(e±) can be m(e±)c2/2 as is necessary. In other words, the charge of the electron must be distributed throughout the interior of the electron, as we assumed. Summing up: The rest mass of the electron and positron originates from the sum of the rest masses of N′/4 electron neutrinos or anti-electron neutri- nos in cubic lattices plus the mass in the energy of N′/2 electric oscillations in the neutrino lattices. That means that neither the electron nor the positron are point particles. The electric oscillations are attached to the neutrinos 9 by the weak nuclear force. The sum of the charge elements of the electric oscillations accounts for the elementary charge of the electron, respectively positron. 2 The spin and magnetic moment of the electron The model of the electron we have proposed in the preceding chapter has, in order to be valid, to pass a crucial test; the model has to explain satisfac- torily the spin and the magnetic moment of the electron. When Uhlenbeck and Goudsmit [8] (U&G) discovered the existence of the spin of the electron they also proposed that the electron has a magnetic moment with a value equal to Bohr’s magnetic moment µB = e¯h/2m(e±)c. Bohr’s magnetic mo- ment results from the motion of an electron on a circular orbit around a proton. The magnetic moment of the electron postulated by U&G has been confirmed experimentally, but has been corrected by about 0.11% for the so-called anomalous magnetic moment. If one tries to explain the magnetic moment of the electron with an electric charge moving on a circular orbit around the particle center, analogous to the magnetic moment of hydrogen, one ends up with velocities larger than the velocity of light, which cannot be, as already noted by U&G. It remains to be explained how the magnetic moment of the electron comes about. We will have to explain the spin of the electron first. The spin, or the intrinsic angular momentum of a particle is, of course, the sum of the angular momentum vectors of all components of the particle. In the electron these are the neutrinos and the electric oscillations. Each neutrino has spin 1/2 and in order for the electron to have s = 1/2 all, or all but one, of the spin vectors of the neutrinos in their lattice must cancel. If the neutrinos are in a simple cubic lattice as in Fig.(1) and the center particle of the lattice is not a neutrino, as in Fig.(1), the spin vectors of all neutrinos in the lattice cancel, Σ j(ni) = 0, provided that the spin vectors of the electron neutrinos of the lattice point in opposite direction at their mirror points in the lattice. Otherwise the spin vectors of the neutrinos would add up and make a very large angular momentum. We follow here the procedure we used in [17] to explain the spin of the muons. The spin vectors", + "of the lattice point in opposite direction at their mirror points in the lattice. Otherwise the spin vectors of the neutrinos would add up and make a very large angular momentum. We follow here the procedure we used in [17] to explain the spin of the muons. The spin vectors of all electron neutrinos in the electron cancel just as the spin vectors of all muon and electron neutrinos 10 in the muons cancel because there is a neutrino vacancy at the center of their lattices, (Fig.(1) of [17]). We will now see whether the electric oscillations in the electron contribute to its angular momentum. As we said in context with Eq.(7) there must be two times as many electric oscillations in the electron lattice than there are neutrinos. The oscillation pairs can either be the two oscillations in a standing wave or they can be two circular oscillations. Both the standing waves and the circular oscillations are non-progressive and can be part of the rest mass of a particle. We will now assume that the electric oscillations are circular. Circular oscillations have an angular momentum ⃗j = m⃗r × ⃗v. And, as in the case of the spin vectors of the neutrinos, all or all but one of the O(109) angular momentum vectors of the electric oscillations must cancel in order for the electron to have spin 1/2. As in [13] we will describe the superposition of the two circular oscillations by x(t) = exp[iωt] + exp[−i(ωt + π)] , (19) y(t) = exp[i(ωt + π/2)] + exp[−i(ωt + 3π/2)] , (20) that means by the superposition of a circular oscillation with the frequency ω and a second circular oscillation with the frequency −ω. The latter oscil- lation is shifted in phase by π. Negative frequencies are permitted solutions of the equations of motion in a cubic lattice, Eqs.(7,13) of [13]. As is well- known oscillating electric charges should emit radiation. However, this rule does already not hold in the hydrogen atom, so we will assume that the rule does not hold in the electron either. In circular oscillations the kinetic energy is always equal to the potential energy and the sum of both is the total energy. From Epot + Ekin = 2 Ekin = Etot (21) follows with Ekin = Θ ω2/2 and Etot = ¯hω that 2 Ekin = Θ ω2 = ¯hω. Θ is the moment of inertia. When we superpose the two circular oscillations with ω and −ω of Eqs.(19,20) we have 2 × 2 Ekin = 2 Θ ω2 = ¯hω , (22) from which follows that the angular momentum is j = Θ ω = ¯h/2 . (23) 11 That means that each of the O(109) pairs of superposed circular oscillations has an angular momentum ¯h/2. The circulation of the oscillation pairs in Eqs.(19,20) is opposite for all ω of opposite sign. It follows from the equation for the displacements un of the lattice points un = Aei(ω t + nφ) , (24) (Eq.(5) in [13]) that the velocities of the lattice points are", + "momentum ¯h/2. The circulation of the oscillation pairs in Eqs.(19,20) is opposite for all ω of opposite sign. It follows from the equation for the displacements un of the lattice points un = Aei(ω t + nφ) , (24) (Eq.(5) in [13]) that the velocities of the lattice points are given by vn = ˙un = i ωn un . (25) The sign of ωn changes with the sign of φ because the frequencies are given by Eq.(13) of [13], that means by ωn = ± ω0 [ φn + φ0 ] . (26) Consequently the circulation of the electric oscillations is opposite to the circulation at the mirror points in the lattice and the angular momentum vectors cancel, but for the angular momentum vector of the electric oscillation at the center of the lattice. The center circular oscillation has, as all other electric oscillations, the angular momentum ¯h/2 as Eq.(23) says. The angular momentum of the entire electron lattice is therefore j(e±) = X j(ni) + X j(eli) = j(el0) = ¯h/2 , (27) as it must be for spin s = 1/2. The explanation of the spin of the electron given here follows the explanation of the spin of the baryons in [13], as well as the explanation of the absence of spin in the mesons. A valid explanation of the spin must be applicable to all particles, in particular to the electron, the prototype of a particle with spin. We will now turn to the magnetic moment of the electron which is known with extraordinary accuracy, µ(e±) = 1.001 159 652 187 µB, according to the Review of Particle Physics [14], with µB being the Bohr magneton. The decimals after 1.00 µB are caused by the anomalous magnetic moment which we will not consider. As is well-known the magnetic dipole moment of a particle with spin is, in Gaussian units, given by ⃗µ = g e¯h 2mc ⃗s , (28) where g is the dimensionless Land´e factor, m the rest mass of the particle and ⃗s the spin vector. The g-factor has been introduced in order to bring the 12 magnetic moment of the electron into agreement with the experimental facts. As U&G postulated and as has been confirmed experimentally the g-factor of the electron is 2. With the spin s = 1/2 of the electron the magnetic dipole moment of the electron is then µ(e±) = e¯h/2m(e±)c , (29) or one Bohr magneton in agreement with the experiments, neglecting the anomalous moment. For a structureless point particle Dirac [9] has explained why g = 2 for the electron. However we consider here an electron with a finite size and which is at rest, which means that the velocity of the center of mass is zero. When it is at rest the electron has still its magnetic moment. Dirac’s theory does therefore not apply here. The only part of Eq.(28) that can be changed in order to explain the g-factor of an electron with structure is the ratio e/m which deals with the spatial distribution of charge and mass.", + "at rest the electron has still its magnetic moment. Dirac’s theory does therefore not apply here. The only part of Eq.(28) that can be changed in order to explain the g-factor of an electron with structure is the ratio e/m which deals with the spatial distribution of charge and mass. In the classical electron models the mass originates from the charge. However that is not necessarily al- ways so. If part of the mass of the electron is non-electrodynamic and the non-electrodynamic part of the mass does not contribute to the magnetic moment of the electron, which to all that we know is true for neutrinos, then the ratio e/m in Eq.(28) is not e/m(e±) in the case of the electron. The elementary charge e certainly remains unchanged, but e/m depends on what fraction of the mass is of electrodynamic origin and what fraction of m is non-electrodynamic, just as the mass of a current loop does not contribute to the magnetic moment of the loop. From the very accurately known values of αf, m(π±)c2 and m(e±)c2 and from Eq.(9) for the energy in the electric oscillations in the electron Eν(e±) = αf/2 · m(π±)c2/2 = 0.996570 m(e±)c2/2 follows that very nearly one half of the mass of the electron is of electric origin, whereas the other half of m(e±) is made of neutrinos which do not contribute to the magnetic moment. That means that in the electron the mass in e/m is practically m(e±)/2. The magnetic moment of the electron is then ⃗µe = g e¯h 2m(e±)/2 · c⃗s , (30) and with s = 1/2 we have µ(e±) = g e¯h/2m(e±)c. Because of Eq.(29) the g-factor must be equal to one and is unnecessary. In other words, if the electron is composed of the neutrino lattice and the electric oscillations as we have suggested, then the electron has the correct magnetic moment µe = e¯h/2m(e±)c, if exactly 1/2 of the electron mass consists of neutrinos. 13 The preceding explanation of the magnetic moment of the electron has to pass a critical test, namely it has to be shown that the same considerations lead to a correct explanation of the magnetic moment of the muon µµ = e¯h/2m(µ±)c, which is about 1/200th of the magnetic moment of the electron but is known with nearly the same accuracy as µe. Both magnetic moments are related through the equation µµ µe = m(e±) m(µ±) = 1 206.768 , (31) as follows from Eq.(28) applied to the electron and muon. This equation agrees with the experimental results to the sixth decimal. The muon has, as the electron, an anomalous magnetic moment of about 0.11 % µµ, which is too small to be considered here. In the standing wave model [13] the muons consist of a lattice of N′/4 muon neutrinos νµ, respectively anti-muon neutrinos ¯νµ, of N′/4 electron neutrinos and the same number of anti-electron neutrinos plus an elemen- tary electric charge. For the explanation of the magnetic moment of the muon we follow the same reasoning we have used for the explanation of", + "a lattice of N′/4 muon neutrinos νµ, respectively anti-muon neutrinos ¯νµ, of N′/4 electron neutrinos and the same number of anti-electron neutrinos plus an elemen- tary electric charge. For the explanation of the magnetic moment of the muon we follow the same reasoning we have used for the explanation of the magnetic moment of the electron. We say that m(µ±) consists of two parts, one part which causes the magnetic moment and another part which does not contribute to the magnetic moment. The part of m(µ±) which causes the magnetic moment must contain circular electric oscillations without which there would be no magnetic moment. It becomes immediately clear from the small mass of the electron neutrinos and from Eq.(5) for the energy of the electric oscillations in the electron that Σ m(νe) and Eν(e±) are too small, as compared to the energy in the rest masses of all neutrinos in the muons, to make up m(µ±)/2. However, the oscillations in the µ± mesons do not follow Eq.(5) for the oscillation energy in the electron, but rather Eq.(4) for the oscillation energy in the muons. Both differ by the factor αf in Eq.(5). But even when the oscillation energy in the muons as given by Eq.(4) is considered, the energy of the electric oscillations in the muons would be only Eν(µ±)/4 = 16.955 MeV, if the electric oscillations are attached to N/4 electron neutrinos, as is the case in the electron. It appears to be necessary to consider the case that the electric oscillations in the µ± mesons are attached to all neutrinos of the electron neutrino type in the µ± lattice, regardless whether they are electron neutrinos or anti- electron neutrinos. That would mean that the electric charge is distributed 14 uniformly in the µ± lattice. There are, as has been shown in the paragraph below Eq.(31) of [13], 3/4·N neutrinos of the electron neutrino type in the muons, of which N/4 neutrinos originate from the charge e± carried by µ±. If the electric oscillations are attached to 3/4·N electron neutrinos, regardless of their type, then the energy in all electric oscilllations or the energy in the electric charge is, with Eq.(8) and Eν(µ±) = Eν(π±) = 67.82 MeV from Eq.(31) in [13], as well as with m(µ±)c2 = 105.6583 MeV, given by 3/4 · Eν(µ±) = 3/4 · 67.82 MeV = 50.865 MeV = 0.4814 m(µ±)c2 ∼= 1/2 · m(µ±)c2 . (32) In other words, the energy in the electric oscillations or the electric charge makes up, in a good approximation, 1/2 of the mass of the muons. The other half of the rest mass of the muons consists of the sum of the rest masses of the neutrinos in the muon lattice plus the oscillation energy of the muon neutrinos, neither of which contributes to the magnetic moment. It is 1/4·Eν(µ±)+N/4·m(νµ)c2+3/4·Nm(νe)c2 = 53.347 MeV = 0.50490 m(µ±)c2 . (33) The theoretical total energy in the rest mass of the muons is then E(m(µ±)) = 0.9863 m(µ±)c2(exp). In simple terms, if Eν(π±) = Eν(µ±) = 1/2·m(π±), not 0.486 m(π±) as in", + "neither of which contributes to the magnetic moment. It is 1/4·Eν(µ±)+N/4·m(νµ)c2+3/4·Nm(νe)c2 = 53.347 MeV = 0.50490 m(µ±)c2 . (33) The theoretical total energy in the rest mass of the muons is then E(m(µ±)) = 0.9863 m(µ±)c2(exp). In simple terms, if Eν(π±) = Eν(µ±) = 1/2·m(π±), not 0.486 m(π±) as in Eq.(27) of [13], then it follows from 3/4·Eν(µ±) = 3/8·m(π±) and from the neutral part of the muon mass in Eq.(33), which is likewise ≈3/8·m(π±), that the rest mass of the muons is m(µ±) ∼= 3/8·m(π±) + 3/8·m(π±) = 3/4· m(π±), as it must be in a first approximation, whereas the actual m(µ±) is 1.00937·3/4·m(π±). That means that in a good approximation the charged part of the rest mass of the muons is 1/2 of the mass of the muons. If the charged part of the muon mass as expressed by Eq.(32) makes up 1/2 of the mass of the muons and if the other part of the muon mass does not contribute to the magnetic moment, then the magnetic moment of the muon is given by ⃗µµ = e¯h 2m(µ±)/2 · c · ⃗s . (34) With s = 1/2 we have µµ = e¯h/2m(µ±)c as it must be, without the artificial g-factor. 15 Conclusions One hundred years of sophisticated theoretical work have made it abundantly clear that the electron is not a purely electromagnetic particle. There must be something else in the electron but electric charge. It is equally clear from the most advanced scattering experiments that the “something else” in the electron must be non-interacting, otherwise it could not be that we find that the radius of the electron must be smaller than 10−16 cm. The only non- interacting matter we know of with certainty are the neutrinos. So it seems to be natural to ask whether neutrinos are not part of the electron. Actually we have not introduced the neutrinos in an axiomatic manner but rather as a consequence of our standing wave model of the stable mesons, baryons and µ-mesons. It follows necessarily from this model that after the decay of the µ−meson there must be electron neutrinos in the emitted electron, and that they make up one half of the mass of the electron. The other half of the energy in the electron originates from the energy of electric oscillations. The theoretical rest mass of the electron agrees, within 1% accuracy, with the experimental value of m(e±). We have learned that the charge of the electron is not concentrated in a single point, but rather is distributed over O(109) elements which are held together with the neutrinos by the weak nuclear force. The sum of the charges in the electric oscillations is, within the accuracy of the parameters, equal to the elementary electrical charge of the electron. From the explanation of the mass and charge of the electron follows, as it must be, the correct spin and magnetic moment of the electron, the other two fundamental features of the electron. With a cubic lattice of anti-electron neutrinos we also arrive with the same considerations as above", + "electron. From the explanation of the mass and charge of the electron follows, as it must be, the correct spin and magnetic moment of the electron, the other two fundamental features of the electron. With a cubic lattice of anti-electron neutrinos we also arrive with the same considerations as above at the correct mass, charge, spin and magnetic moment of the positron. Acknowledgements Contributions of Professor J. Zierep and of Dr. T. Koschmieder are gratefully acknowledged. References [1] Thomson, J.J. Phil.Mag. 44,293 (1897). 16 [2] Lorentz, H.A. Enzykl.Math.Wiss. Vol.5,188 (1903). [3] Poincar´e, H. Compt.Rend. 140,1504 (1905). Translated in: Logunov, A.A. On The Articles by Henri Poincare “On The Dynamics of the Electron”, Dubna JINR (2001). [4] Ehrenfest, P. Ann.Phys. 24,204 (1907). [5] Einstein, A. Sitzungsber.Preuss.Akad.Wiss. 20,349 (1919). [6] Pauli, W. Relativit¨atstheorie, B.G. Teubner (1921). Translated in: Theory of Relativity, Pergamon Press (1958). [7] Poincar´e, H. Rend.Circ.Mat.Palermo 21,129 (1906). [8] Uhlenbeck, G.E. and Goudsmit, S. Naturwiss. 13,953 (1925). [9] Dirac, P.A.M. Proc.Roy.Soc.London A117,610 (1928). [10] Gottfried, K. and Weisskopf, V.F. Concepts of Particle Physics, Vol.1, p.38. Oxford University Press (1984). [11] Schr¨odinger, E. Sitzungsber.Preuss.Akad.Wiss. 24,418 (1930). [12] Mac Gregor, M.H. The Enigmatic Electron, Kluwer (1992). [13] Koschmieder, E.L. http://arXiv.org/phys/0602037 (2006). [14] Eidelman, S. et al. Phys.Lett.B 592,1 (2004). [15] Born, M. and v. Karman, Th. Phys.Z. 13,297 (1912). [16] Nambu, Y. Prog.Th.Phys. 7,595 (1952). [17] Koschmieder, E.L. http://arXiv.org/physics/0308069 (2003), Muons: New Research, Nova (2005). 17", + "arXiv:2006.13896v2 [physics.chem-ph] 20 Jul 2020 Photoisomerization-coupled electron transfer Jakub K. Sowa,1, a) Emily A. Weiss,1 and Tamar Seideman1, b) Department of Chemistry, Northwestern University, Evanston, IL 60208, USA. Photochromic molecular structures constitute a unique platform for constructing molecular switches, sensors and memory devices. One of their most promising applications is as light-switchable electron acceptor or donor units. Here, we investigate a previously unexplored process that we postulate may occur in such systems: an ultrafast electron transfer triggered by a simultaneous photoisomerization of the donor or the acceptor moiety. We propose a theoretical model for this phenomenon and, with the aid of DFT calculations, apply it to the case of a dihydropyrene-type photochromic molecular donor. By considering the wavepacket dynamics and the photoisomerization yield, we show that the two processes involved, electron transfer and photoisomerization, are in general inseparable and need to be treated in a unified manner. We finish by discussing how the efficiency of photoisomerization-coupled electron transfer can be controlled experimentally. I. INTRODUCTION The phenomenon of photoisomerization, in which a molecule undergoes a structural change following its ex- citation with light, not only plays a critical role in the hu- man visual perception1–3 but has also attracted attention in the field of nanotechnology where it can potentially be used to construct molecular switches, engines and opti- cal memory devices.4–7 There now exist several families of chemical compounds known to undergo reversible photoi- somerization which typically involves either a ring clos- ing/opening reaction or a cis-trans isomerization.8 Pho- toisomerization of such organic compounds generally oc- curs through a motion of the photo-excited vibrational wavepacket through a conical intersection (CI), followed by an eventual vibrational relaxation.2,9–12 The mechanism of photoisomerization in organic molecules has been extensively studied over the past three decades and several common types of theoretical approaches can be identified. The first class of methods entails using (often high-level CASPT2/CASSCF) ab ini- tio calculations to construct the potential energy surface around the two minima (corresponding to the two pho- toisomers) and the conical intersection located between them,13–15 although ab initio molecular dynamics simula- tions have also been used to study the non-adiabatic pho- toisomerization dynamics.16–19 Another approach, which will also be used in this work, involves modelling the pho- toisomerization dynamics using idealised potential en- ergy surfaces often also accounting for the dissipative ef- fects arising from interactions with the broader molecular environment.20–25 From both a fundamental and a technological per- spective, one of the most exciting applications of pho- tochromic compounds is as light-switchable electron ac- ceptors or donors.26 That is, due to the difference in oxi- dation (or reduction) potentials between the two isomers, photochromic compounds may act as efficient electron donors (or acceptors) only in one of their isomeric forms. a)Electronic mail: jakub.sowa@northwestern.edu b)Electronic mail: t-seideman@northwestern.edu It has been suggested that such systems can form the basis of molecular optical switches and memory devices, and have therefore attracted considerable experimental interest.27–35 Here, we suggest that photochromic electron donors (or acceptors) can also give rise to a novel phenomenon termed photoisomerization-coupled electron transfer. To illustrate this concept, let us", + "been suggested that such systems can form the basis of molecular optical switches and memory devices, and have therefore attracted considerable experimental interest.27–35 Here, we suggest that photochromic electron donors (or acceptors) can also give rise to a novel phenomenon termed photoisomerization-coupled electron transfer. To illustrate this concept, let us consider a photochromic unit which in one of its isomeric forms can act as an ef- ficient electron donor, and which is connected to a (non- photochromic) acceptor unit, as schematically shown in Fig. 1(a). Suppose that the photochromic donor unit was initially prepared in the isomeric form with a high oxidation potential [denoted as A in Fig. 1(a)] so that the electron transfer is not possible. One can then use a laser pulse of an appropriate wavelength to excite [and thus photoisomerise to what is denoted as isomer B in Fig. 1(a)] the photochromic donor unit and hence enable the electron transfer to take place. In the case of weak coupling between the donor and acceptor, the photoi- somerization and the electron transfer should occur on very different timescales. First, a laser pulse excites the photochromic unit and induces its photoisomerization. Only later, the slow (non-adiabatic) electron transfer, from what is now a suitable electron donor, takes place. In general, however, the two processes are not separable. This is especially important if the donor and acceptor units are bound covalently – this is expected to give rise to relatively strong electronic coupling and thus a fast electron transfer, occurring on a timescale comparable to that of photoisomerization. This simultaneous electron transfer-photoisomerization process, in which the isomer- ization driving the electron transfer will itself be affected by the charge transfer dynamics, will be the focus of this work. As stated above, photochromic electron acceptor and donor systems have been postulated as a possible ba- sis for various optoelectronic applications such as single- molecule memory devices enabling a non-destructive readout.30,32 Such devices can, in principle, operate on ultrafast timescales and it is therefore important to understand the dynamical processes which can occur in these systems. Photoisomerization-coupled electron transfer could furthermore unlock molecular-switching 2 functionalities not accessible in the case of ‘conventional’ photo-switching dynamics (for which the electron trans- fer and photoisomerization take place sequentially). To the best of our knowledge, neither the dynamics nor the mechanism of photoisomerization-coupled electron transfer have been studied to date. In this work, our pri- mary aim is to develop a theoretical description and build an intuitive understanding of this phenomenon. To this end, in the latter part of this work we will consider an il- lustrative example of a dihydropyrene-type photochromic electron donor. e- h \u0001 electron acceptor isomer A (donor) electron acceptor e- qt qc Energy |SB |SA |CTB |CTA qt Energy \u0000CTA \u0002SA \u0003CTB \u0004SB |SA |SB |CTB |CTA isomer B (donor) (a) (b) (c) FIG. 1. (a) Schematic illustration of electron transfer induced by a photoisomerization of a donor unit. (b) Potential energy surfaces for the neutral (singlet) and charge-transfer states. For clarity, J was set to zero and the charge-transfer states in (b)", + "|SA |SB |CTB |CTA isomer B (donor) (a) (b) (c) FIG. 1. (a) Schematic illustration of electron transfer induced by a photoisomerization of a donor unit. (b) Potential energy surfaces for the neutral (singlet) and charge-transfer states. For clarity, J was set to zero and the charge-transfer states in (b) were shifted downwards. (c) Diabatic states plotted at qc = 0. CI positions are denoted above. II. THEORETICAL MODEL We assemble a minimal (but therefore also a general) description of the problem at hand. Our system com- prises a photochromic donor (acceptor) unit electroni- cally coupled to an ordinary acceptor (donor) moiety. We assume that the photochromic unit can exist in either of its photoisomeric forms, henceforth referred to as pho- toisomers A and B, and in either its neutral or cationic (anionic) charge state. There exist therefore four rele- vant electronic states: two singlet states describing neu- tral donor and acceptor units, |SA⟩and |SB⟩, and two charge-transfer states (describing the cationic donor and anionic acceptor units), |CTA⟩and |CTB⟩. We assume that (de-)charging of the non-photochromic unit is not accompanied by a significant displacement of its nuclear coordinates. This should be a reasonable assumption es- pecially if that moiety corresponds to a semiconductor quantum dot.27 As stated before, the photoisomerization of the pho- tochromic unit involves a motion of a photoexcited wavepacket through a CI. The minimal model for the neutral photochromic moiety comprises therefore two di- abatic singlet states (corresponding to isomers A and B) coupled to the so-called tuning and coupling vibrational modes.36 The Hamiltonian for the neutral states is there- fore (ℏ= 1):12,21,36 H0 = X i,i′=SA,SB |i⟩hiδi,i′ + λ0 qc(1 −δi,i′)⟨i′| , (1) where hi = εi + κ(i) t qt + T . In the above, qt and qc are the coordinates of the tuning and coupling mode, respectively, εi is the energy of state i (at the tuning vibrational coordinate qt = 0), κ(i) t is the coupling con- stant of the i-state to the tuning mode, and λ0 deter- mines the coupling between the diabatic states SA and SB. The minimum of each of the diabatic states is given by ¯εi = εi − \u0010 κ(i) t /√2ωt \u00112 with a corresponding dimen- sionless displacement: ∆i = −κ(i) t /ωt. Finally, T is the total vibrational operator: T = X k=c,t ωk 2 \u0012 −∂2 ∂q2 k + q2 k \u0013 , (2) where ωk is the frequency of the mode k. We assumed that both the tuning and the coupling mode can be de- scribed using harmonic potentials. In reality, the tuning mode (which acts here as the reaction coordinate) may be expected to exhibit a large degree of anharmonicity. Nonetheless, we expect the model presented here to cap- ture the essential physics of the considered phenomenon. A similar potential energy surface (including a conical intersection) can also be constructed for open-shell pho- tochromic systems.37,38 The Hamiltonian for the charge- transfer states (with a charged photochromic unit) can therefore similarly be written as: H1 = X j,j′=CTA,CTB |j⟩hjδj,j′ + λ1 qc(1 −δj,j′)⟨j′| , (3)", + "the considered phenomenon. A similar potential energy surface (including a conical intersection) can also be constructed for open-shell pho- tochromic systems.37,38 The Hamiltonian for the charge- transfer states (with a charged photochromic unit) can therefore similarly be written as: H1 = X j,j′=CTA,CTB |j⟩hjδj,j′ + λ1 qc(1 −δj,j′)⟨j′| , (3) with hj = εj + κ(j) t qt + κ(j) c qc + T . Crucially, as it will be demonstrated below and is also shown in the supple- mentary material, the ground states of the charged pho- toisomers are in general displaced with respect to their neutral counterparts. The neutral and charge-transfer states are coupled 3 electronically via H0−1 = X i,j Jij(|i⟩⟨j| + |j⟩⟨i|) , (4) where |i⟩= {|SA⟩, |SB⟩}, |j⟩= {|CTA⟩, |CTB⟩}, and Jij is the strength of the coupling between the i and j states. An example of the four diabatic states considered here is plotted (as a function of the nuclear coordinates qt and qc) in Fig. 1. The energy differences between the two photoisomers (i.e. between ¯εSA and ¯εSB, and between ¯εCTA and ¯εCTB) are determined by the nature of the pho- tochromic unit, see supplementary material for several examples. On the other hand, the energy differences be- tween the neutral singlet and charge-transfer states can be tuned by choosing an appropriate (non-photochromic) donor or acceptor unit. Finally, both vibrational modes are coupled to their (separate) wider environments which are modelled as col- lections of harmonic oscillators,39,40 see supplementary material. They account for the solvent environment of the donor-acceptor system as well as, to a lesser degree, the residual intramolecular vibrational modes. These in- teractions induce vibrational relaxation (damping of the wavepacket at the rates γk, k = c, t) which will be mod- elled here using the diabatic damping approximation41–43 (DDA) within the Redfield theory which itself relies on the Born and Markov approximations justified in the limit of weak environmental coupling.44 It will be there- fore assumed that the phononic environments remain thermalised at all times and can be described using con- tinuous spectral densities. We note that DDA has pre- viously been used in the modelling of ultrafast electron transfer where it compared favourably with the secular approximation method.45 Furthermore, unlike Redfield theory itself, DDA does not violate positivity nor does it require a detailed knowledge of the vibrational environ- ment. Given the limitations of the (phenemenological) weak-coupling approach used in this work, extending the theory discussed here to include more sophisticated21 or even numerically exact46 descriptions of environmental interactions would be an interesting, and important, av- enue for future research. We nonetheless believe that the results presented in this work grasp the fundamental physics of the considered phenomenon. In what follows, we will also disregard any direct coupling between the electronic degrees of freedom and the outer-sphere envi- ronment which plays a lesser role in the case of non-polar solvents (as used, for instance, in Ref. 27). Unless specified otherwise, we will henceforth assume that the initial state was prepared by an ultrashort laser pulse which induces a vertical excitation of the", + "of freedom and the outer-sphere envi- ronment which plays a lesser role in the case of non-polar solvents (as used, for instance, in Ref. 27). Unless specified otherwise, we will henceforth assume that the initial state was prepared by an ultrashort laser pulse which induces a vertical excitation of the neutral photoisomer A, Fig. 1(c). The initial density matrix is therefore ρ(0) = |SB⟩|0⟩⟨0|⟨SB| , (5) where |0⟩denotes the vibrational ground state of the di- abatic state |SA⟩. 0 0.5 1 Population |SA |SB |CTA |CTB 0 500 1000 Time (fs) 0 0.5 1 Population (a) A B acceptor hν e- CN NC CN NC acceptor (b) (c) 0 500 1000 Time (fs) FIG. 2. (a) Schematic of the studied model system. (b, c) Populations of the diabatic states as a function of time in the (b) absence, and (c) presence of vibrational damping γk = 2 meV and T = 300 K. III. RESULTS AND DISCUSSION A. Model system We demonstrate the principle of photoisomerization- coupled electron transfer on a concrete example although most of our conclusions are general. We consider a system akin to that studied by Liddell et al.33 which comprises a ‘conventional’ acceptor unit47 and a pho- tochromic dihydropyrene-type molecule acting as a (po- tential) electron donor.48 As shown in Fig. 2(a), the two possible photoisomers of the donor unit (A and B) pos- sess different oxidation potentials so that photoisomeris- tion of the donor can be used to control the electron transfer. We note that the considered photochromic unit belongs to the C2h point group in both of its neutral iso- meric forms. We do not consider any linking groups in our calculations in order to make full use of the molecular symmetry. We parameterise our (minimal and prototypical) model as follows. First, geometry relaxation of the neu- tral and cationic forms of the two photoisomers is per- formed to obtain the energy differences between the two isomers in their respective charge states (all calculations are performed in NWChem at the B3LYP/6-311+G** level of theory49). The relative alignment of the neu- tral and charge transfer states is chosen such that the ground states of |CTA⟩and |CTB⟩lie above and below those of the |SA⟩and |SB⟩states, respectively: ¯εSA = 0, ¯εSB = −0.42, ¯εCTA = 0.32, and ¯εCTB = −0.5 (all in eV). 4 Symmetry analysis reveals that the vibrational modes are Γvib = 24Ag + 18Bg + 19Au + 23Bu. The totally- symmetric Ag modes can be easily identified as the tun- ing modes,36 whereas, as discussed in the supplementary material, the coupling modes are most likely of Bu sym- metry. Since the electronic ground states of A, B and B+ are all of Ag symmetry, we indeed find that only the Ag modes are displaced between the A and B/B+ isomers. Meanwhile, the A+ structure distorts into the C2 symmetry which is accompanied by a displacement of the Ag and Au modes. Consequently, no displacement of the coupling mode is associated with the electron transfer from the photochromic unit, κ(j) c = 0. In order to deter-", + "A and B/B+ isomers. Meanwhile, the A+ structure distorts into the C2 symmetry which is accompanied by a displacement of the Ag and Au modes. Consequently, no displacement of the coupling mode is associated with the electron transfer from the photochromic unit, κ(j) c = 0. In order to deter- mine the displacement parameters κ(i) t , we use the neu- tral A structure as the reference geometry and follow a Dushinsky-type procedure as outlined in the supplemen- tary material. We choose the Ag mode with the largest displacement for the A →B transition to act as the tun- ing mode in our calculation. This mode corresponds to the stretching/compressing of the carbon-carbon bond formed during the photoisomerization (see supplemen- tary material) and has a frequency of ωt = 39.5 meV. The relative displacements are κ(SA) t = −0.235, κ(SB) t = 0.235, κ(CTA) t = −0.217, and κ(CTB) t = 0.217 (all in eV). The resulting alignment of the energy levels is very similar to that shown in Fig. 1(c), see supplementary material. Crucially, therefore, the charge transfer is accompanied by a displacement of the tuning (isomerization) coordi- nate. We do not attempt here to extract the interstate cou- plings λi (or ωc) from ab initio calculations, cf. Ref. 50, resorting instead to values previously suggested in the literature,51 ωc = 0.112 eV and λ0 = λ1 = 49.6 meV, as- suming for simplicity that inter-state coupling is identical in the neutral and charge-transfer states. To further re- duce the number of parameters, we set Jij = J = 5 meV, unless stated otherwise. B. Properties of the model Let us begin by exploring the properties of our model system. (i) Firstly, in the absence of the initial photoex- citation, that is, when ρ(0) = |SA⟩|0⟩⟨0|⟨SA|, the elec- tron transfer is not energetically possible as the |CTA⟩ state lies above the |SA⟩state and there exists an en- ergy barrier for the direct isomerization. Consequently, on the considered timescale, the system essentially re- mains in the (meta-stable) |SA⟩state, see supplementary material. (ii) In the absence of donor-acceptor electronic coupling, that is for J = 0, following a photoexciation of the A isomer the molecular system simply isomerises from A to B with a yield of roughly 0.74 (for γk = 2 meV and temperature T = 300 K). As explicitly shown in the supplementary material, this process takes place within less than 1 ps. (iii) For the system initially found in the vibrational ground state of state |SB⟩, the electron -15 -10 -5 0 5 qt 0 0.1 0.2 0.3 -10 -5 0 5 qt 0 0.2 0.4 0.6 (a) (b) (c) (\u0005 \u0006 t = 500 fs \u0007k = 0 meV J = 0 meV J = 5 meV t = 500 fs \bk = 2 meV FIG. 3. (a, b) Projections of the wavepacket on the tuning coordinate as a function of time for (a) γk = 0, and (b) γk = 2 meV. (c, d) Projections of the wavepacket on the tuning coordinate after 500 fs of wavepacket evolution", + "500 fs \bk = 2 meV FIG. 3. (a, b) Projections of the wavepacket on the tuning coordinate as a function of time for (a) γk = 0, and (b) γk = 2 meV. (c, d) Projections of the wavepacket on the tuning coordinate after 500 fs of wavepacket evolution for (c) γk = 0, and (d) γk = 2 meV; T = 300 K. transfer to the state |CTB⟩takes place without the need for laser excitation due to the relatively small energy bar- rier between the |SB⟩and |CTB⟩states, see Fig. 1(c) and the supplementary material. (iv) Finally, as we shall dis- cuss in the remainder of this work, the photoexcitation of the wavepacket located in the ground state of the |SA⟩ state triggers a simultaneous isomerization of the pho- tochromic donor moiety and an electron transfer from that donor to the acceptor unit. C. Dynamics We first consider the fully unitary dynamics, that is, we set γk = 0. As shown in Fig. 2(b), following the ver- tical photoexcitation, we observe coherent oscillations of the populations of the four diabatic states in which the small initial rise of the population of the |SA⟩state is fol- lowed by a fast increase of the charge-transfer states. As shown in Fig. 2(c), in the presence of vibrational dissipa- tion the initial coherent oscillations are largely damped which is accompanied by a more efficient electron trans- fer. The photoisomerization takes place within roughly 600 fs. This is accompanied by the levelling offof the populations of the |SA⟩state (formed by a propagation of the vibrational wavepacket through the conical inter- section). In the case of a successful photoisomerization, the ‘excess’ electron is at longer times delocalised over the non-photochromic acceptor and the donor unit (in its B form), see the supplementary material for the population dynamics at longer times. This occurs due to relatively strong donor-acceptor coupling that is comparable to the energy difference between the |SB⟩and |CTB⟩states in 5 (a) (b) 0 5 10 15 Donor-acceptor coupling, J (meV) 0.65 0.7 0.75 0.8 0.85 Isomerisation yield, 0 0.5 1 1.5 2 Time (ps) 0 0.5 1 Population |SA |SB |CTA |CTB FIG. 4. (a) Isomerization yield Φ as a function of the donor-acceptor coupling, J. (b) Diabatic population dynamics for very weak donor-acceptor coupling, J = 0.4 meV. Remaining parameters as in Fig. 2(c). our model system (and is not therefore universally true). Surprisingly, we also observe a small (but non-negligible) population of the |CTA⟩state. This corresponds to the electron transfer having taken place despite an unsuccess- ful photoisomerization of the donor unit. Generally, this has a chance of occurring as long as the photoinduced wavepacket propagates through the (avoided) crossing of the |SB⟩and |CTA⟩diabatic states, as it can be inferred from Fig. 1(c). We next demonstrate the inseparability of the photoi- somerization and electron transfer processes. First, in Figs. 3(a, b), we plot the projection of the wavepacket in the diabatic state |SB⟩on the tuning coordinate as a function of time. We observe the expected oscillations of the wavepacket, damped towards qt = ∆SB", + "next demonstrate the inseparability of the photoi- somerization and electron transfer processes. First, in Figs. 3(a, b), we plot the projection of the wavepacket in the diabatic state |SB⟩on the tuning coordinate as a function of time. We observe the expected oscillations of the wavepacket, damped towards qt = ∆SB in the case of non-zero γt. Figs. 3(c) and (d) show the projection of the wavepacket at time t = 500 fs for two values of the donor- acceptor coupling: J = 0 and 5 meV. Crucially, the pos- sibility of the electron transfer (non-zero J) affects the shape as well as the magnitude of the wavepacket. The shape of the wavepacket (after certain propagation time) depends non-trivially on the details of the potential en- ergy surface. The decrease in its magnitude, on the other hand, can be easily understood as a consequence of pop- ulating the charge transfer states. It is also useful to consider the yield of photoiso- merization which one can define (only in the presence of vibrational damping) as Φ = PSB(tss) + PCTB(tss) where Pi(t) = Tr [|i⟩⟨i|ρ(t)] and tss is the time at which the dynamics of the system reaches its quasi-steady- state.21 For the parameters used here tss ∼3 ps (see supplementary material for the population dynamics for the considered values of the donor-acceptor coupling). Fig. 4(a) shows the photoisomerization yield as a func- tion of the donor-acceptor coupling, J. The initial in- crease of Φ with increasing J is due to the possibility of charge transfer which leads to populating the (ener- getically more stable) |CTB⟩state. On the other hand, for larger values of the donor-acceptor coupling, the pho- toisomerization yield marginally decreases which can be attributed to the formation of the |CTA⟩products. Cru- cially, however, Fig. 4(a) reveals that the photoisomeriza- tion yield depends on the strength of the donor-acceptor coupling. For the parameters (and the model system) considered here, the changes of Φ with J are relatively modest. It should nevertheless be possible to detect them experimentally, see Ref. 52. Since the possibility of charge transfer from the photochromic donor unit af- fects the overall photoisomerization yield, the two pro- cesses clearly need to be treated within a unified theory, as asserted above. Finally, we turn to the issue of controlling the effi- ciency of the photoisomerization-coupled electron trans- fer. As expected, the rate of the considered electron transfer decreases with decreasing donor-acceptor cou- pling J. As discussed above and shown in Fig. 4(b), in the case of significantly smaller donor-acceptor cou- pling, the photoisomerization and electron transfer take place on very different timescales. The electron transfer (which can be observed as the rise of the populations of the charge transfer states) takes place largely after the photoisomerization is complete (that is, when the |SA⟩ and |SB⟩populations have almost reached their quasi- steady-state values). Consequently, the electron transfer and photoisomerization may essentially be treated as sep- arate processes. It is also worth noting that in the case of much weaker donor-acceptor coupling, no significant populations of the |CTA⟩state are observed, in contrast", + "when the |SA⟩ and |SB⟩populations have almost reached their quasi- steady-state values). Consequently, the electron transfer and photoisomerization may essentially be treated as sep- arate processes. It is also worth noting that in the case of much weaker donor-acceptor coupling, no significant populations of the |CTA⟩state are observed, in contrast to what was discussed above. It is important to note here that the donor-acceptor coupling can be tuned (or at least varied) experimentally by changing the chemical group linking the donor and acceptor units. Although the wavepacket dynamics are sensitive to the details of the potential energy surfaces the latter are in- herent to the considered photochromic system. Similarly, environmental interactions strongly influence the dy- namics of the photoisomerization-coupled electron trans- fer, but are generally difficult to control experimentally. Nonetheless, the efficiency of the electron transfer can be to some extent controlled by changing the offset between the neutral and charge-transfer states. In practice this can be achieved by varying the non-photochromic accep- tor (or, if appropriate, donor) moiety. As can be inferred from Fig. 1(c), by changing the relative energies of the 6 neutral and the charge transfer states, one can modify the energy barrier for the electron transfer for the isomer B. In particular, as we demonstrate in the supplemen- tary material, stabilising the charge transfer states (that is, lowering their energy relative to that of neutral sin- glet states) can push the |SB⟩→|CTB⟩electron transfer into a deep inverted region thus impairing the efficiency of this process. In the context of controlling the dynam- ics of photoisomerization-coupled electron transfer, the use of colloidal quantum dots as the non-photochromic acceptor (or donor) units27 is therefore particularly at- tractive due to the apparent tunability of these systems. (It is possible to shift the positions of the quantum-dot electronic energy levels by simply changing the size of these particles.) IV. CONCLUSIONS In this work, we have introduced and devel- oped a conceptual understanding of the phenomenon of photoisomerization-coupled electron transfer which amounts to a simultaneous and inseparable photoiso- mersation and electron transfer from (or onto) a pho- tochromic donor (or acceptor) unit. We have demon- strated, using the example of a dihydropyrene-type molecular structure, how this process can be modelled and, to some extent, controlled by the chemical design of the donor-acceptor system. We have also shown that this phenomenon can yield products inaccessible through a sequential isomerization and charge transfer reactions. Furthermore, we discussed how the photoisomerization yield can be affected by the possibility of an electron transfer. If a stronger dependence of the photoisomer- ization yield on the donor-acceptor coupling (than ob- served in our prototypical model system) can be re- alised, the process considered here may open an attrac- tive new way of controlling the outcomes of photoisomer- ization reactions. An understanding of the dynamics of photoisomerization-coupled electron transfer should also prove crucial for the future design of molecular switches and memory devices based on the phenomenon consid- ered here, especially when an ultrafast operation of such devices is desirable. While awaiting experimental studies of the consid- ered phenomenon, several", + "An understanding of the dynamics of photoisomerization-coupled electron transfer should also prove crucial for the future design of molecular switches and memory devices based on the phenomenon consid- ered here, especially when an ultrafast operation of such devices is desirable. While awaiting experimental studies of the consid- ered phenomenon, several aspects of photoisomerization- coupled electron transfer can be further explored theoret- ically. These include extending the theory presented here to the multi-mode case, anharmonic potential energy sur- faces and cis-trans isomerization as well as identifying molecular structures suitable for experimental investiga- tions. Designing systems in which photoisomerization and electron transfer are maximally intertwined would be especially important for the future empirical research into the process considered here. It would likewise be interesting to investigate in more detail the role of en- vironmental interactions (also beyond the weak-coupling approximation) including the importance of possible non- Markovian effects. Finally, ab initio molecular dynamics simulations could also be used to study the phenomenon in question. SUPPLEMENTARY MATERIAL See supplementary material for several further exam- ples of photochromic units, details of the theoretical methods, and supporting dynamics calculations. ACKNOWLEDGMENTS The authors thank Roel Tempelaar for comments on the manuscript. This work was supported by the National Science Foundations MRSEC program (DMR-1720139) at the Materials Research Center of Northwestern University. This research was also supported in part through the computational resources and staffcontributions provided for the Quest high per- formance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. DATA AVAILABILITY STATEMENT The data that support the findings of this study are available within the article and its supplementary mate- rial. 1T. Yoshizawa and G. Wald, Nature 197, 1279 (1963). 2D. Polli, P. Alto`e, O. Weingart, K. M. Spillane, C. Manzoni, D. 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Roulleau1, 1 Nanoelectronics Group, SPEC, CEA, CNRS, Universit´e Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France Received XXXX, revised XXXX, accepted XXXX Published online XXXX Key words: single electron source, quantum shot noise, levitons ∗Corresponding author: e-mail christian.glattli@cea.fr Single electron sources enable electron quantum optics experiments where single electrons emitted in a ballis- tic electronic interferometer plays the role of a single photons emitted in an optical medium in Quantum Op- tics. A qualitative step has been made with the recent generation of single charge levitons obtained by apply- ing Lorentzian voltage pulse on the contact of the quan- tum conductor. Simple to realize and operate, the source emits electrons in the form of striking minimal excitation states called levitons. We review the striking properties of levitons and their possible applications in quantum physics to electron interferometry and entanglement. t Schematic generation of time resolved single charges called levitons using Lorentzian voltage pulses applied on a contact. A Quantum Point Contact is used to partition the levitons for further analysis. Injecting levitons on opposite contacts with a delay τ enables to probe electronic like Hong Ou Mandel cor- relations. Copyright line will be provided by the publisher 1 Single electron sources In this introduction, we will distinguish single charge sources from coherent sin- gle electrons sources. The former have been developed for quantum metrology where the goal is to transfer an integer charge at high frequency f through a conductor with good accuracy to realize a quantized current source whose cur- rent I = ef shows metrological accuracy. The latter, the coherent single electrons source, aims at emitting (inject- ing) a single electron whose wave-function is well defined and controlled to realize further single electron coherent manipulation via quantum gates. The gates are provided by electronic beam-splitters made with Quantum Point Con- tacts or provided by electronic Mach-Zehnder and Fabry- Prot interferometers. Here it is important that the injected single electron is the only excitation created in the conduc- tor. The frequency f of injection is not chosen to have a large current, as current accuracy is not the goal, but only to get sufficient statistics on the electron transfer events to extract physical information. 1.1 single charge sources for current standards The first manipulation of single charges trace back to the early 90’s where physicists took advantage of charge quan- tization of a submicronic metallic island nearly isolated from leads by tunnel barriers. The finite energy EC = e2/2C to charge the small capacitor C with a single charge being larger than temperature (typically one kelvin for Copyright line will be provided by the publisher 2 D.C. Glattli et al.: levitons C =1fF), Coulomb Blockade of tunneling occurs allow- ing for single charge manipulations. An architecture made of several island in series lead to single electron pumps and turnstiles [1,2]. Pumps and turnstiles were dedicated to quantum metrology with the hope to realize a current stan- dard where the current I = ef is related to", + "tunneling occurs allow- ing for single charge manipulations. An architecture made of several island in series lead to single electron pumps and turnstiles [1,2]. Pumps and turnstiles were dedicated to quantum metrology with the hope to realize a current stan- dard where the current I = ef is related to the frequency of single charge transfer through the device. A 7-junction pumps lead soon to 15.10−8 current accuracy but limited to too low currents [3] for metrology. Later hybrid super- conducting normal junctions single electron pumps lead to higher operating frequencies to reach 10−6 accuracy [4]. In parallel, electron pumps have been realized using 2D electrons confined in GaAs/GaAlAs semiconductors heterojunctions where the metallic island is replaced by a quantum dot obtained by surface gate depletion. Here pairs of split gates were used to form a narrow constric- tion which selects the number of electronic modes trans- mitted between the dot and the leads and ultimately leads to a 1D tunnel barrier. This Quantum Point Contact (QPC) offers tunability of the tunnel barrier whose fast manip- ulation was exploited to capture a single electron in the dot from the left contact and release it to the right con- tact providing fast accurate electron pumping [5]. Similar double modulated barrier pumps at GHz frequencies lead to average pump current of several hundred picaoamperes [6] more suitable for metrology. Single gate pumps have been implemented using silicon nanowires [7] and etched GaAl/GaAsAl nanowires. Recent improvements gave ac- curacy better than 1.2.10−6 for a 150pA generated current [8] making these devices promising for current standard in quantum metrology while similar devices combined with quantum Hall effect may give a short path to metrology triangle closure [9]. Another approach to single charge sources is based on moving quantum dots carrying integer charges using of Surface Acoustic Waves (SAWs) [10]. In a piezo-electric material like GaAs, a mechanical wave can be generated by the electric field created by interdigitated gates deposited on top of the device. An electrical potential well moving at the surface phonon velocity, a few km/s, can trap elec- trons confined in a 1D wire and the moving quantum dot so realized can transfer them through the conductor at GHz frequencies leading to good current quantization accuracy [11]. 1.2 single electron sources for electron quantum optics In all the examples cited above, the goal was a good accuracy in the charge transfer. In the following we will focus on quantum experiments appropriate for elec- tron quantum optics where the emitted single electrons play the role of flying charge qubits [12]. Depending on the system used the charge or the spin may code a binary qubit information. The flying qubit approach is fundamen- tally different from the standard qubit realization which are based on a static localized two-level system (nuclear or electron spin, atom, charge state of a superconducting metallic island, photonic state of resonator circuits, etc.). Here the information is coded by the presence or not of a particle (electron, photon) propagating or delocalized in spatial modes. Experiments in this direction using the SAW", + "static localized two-level system (nuclear or electron spin, atom, charge state of a superconducting metallic island, photonic state of resonator circuits, etc.). Here the information is coded by the presence or not of a particle (electron, photon) propagating or delocalized in spatial modes. Experiments in this direction using the SAW technique described above have been done recently. SAW assisted transfer of single electrons through depleted 1D channels have demonstrated single electron transfer over several µm between two distant quantum dots [13,14] and the spin state of one and two electrons have been transferred [15]. Using quantum dot sources, single electrons have been emitted at high energy above the conductor Fermi level. The energy and time properties of the wavepacket have been analyzed [16]. The partitioning of spin entangled electron pairs emitted that way has been recently realized [17]. These examples are promising for further quantum experiments where only the spin coherence of electrons is concerned, but regarding orbital quantum coherence, ap- propriate for charge flying qubits, different approaches are needed. The first coherent single electron source has been pro- posed and realized by one of the present authors with the ENS Paris team [18]. It is based on a mesocopic capac- itor which has been precedently realized [19] to check the universal quantization of the charge relaxation resis- tance (called B¨uttiker’s resistance h/2e2) predicted by M. B¨uttiker [20]. Here no dc current is produced but only an ac current made from the periodic injection of single electrons followed by single holes above and below the Fermi energy EF of the target quantum conductor. For ease of operation and further use in electron quantum optics, the conductor is in the integer quantum Hall effect regime where a strong magnetic field quantizes the electron cyclotron orbits and only 1D chiral modes propagating along the sample edges, called edge channels, ensure electron conduction. Here, the capacitor is made so small that energy level quantization makes it behave as a quantum dot. To ensure energy level and charge quantization, the capacitor is weakly connected to the leads, the chiral edge channels, via a quantum point contact which controls the tunnel coupling. The operating principle is as follows, see Fig.1. Starting from a situation where the last occupied energy level is below the Fermi energy, a sudden rise of the voltage applied on a top gate capacitively coupled to the mesoscopic capacitor rises the occupied energy level above the Fermi energy. After a time equal to the life time ≃¯h/D∆of the energy level and con- trolled by the barrier transmission D, an electron is emit- ted at a tunable energy εe above the Fermi level (∆is the energy level spacing). Then restoring the top gate voltage to its initial value pulls down the energy level below the Fermi energy: an electron is captured or equivalently a hole is emitted with a definite energy −εh below the lead Fermi energy. The mesoscopic capacitor single electron source has been used for electron quantum optics experiments like the partitioning of single electrons in an electronic beam splitter [21], an Hanbury", + "Fermi energy: an electron is captured or equivalently a hole is emitted with a definite energy −εh below the lead Fermi energy. The mesoscopic capacitor single electron source has been used for electron quantum optics experiments like the partitioning of single electrons in an electronic beam splitter [21], an Hanbury Brown-Twiss experimental set- Copyright line will be provided by the publisher pss header will be provided by the publisher 3 up commonly used in optics, and the Hong Ou mandel (HOM) electron interferometry where two electron emit- ted with a controlled relative time delay are sent to an elec- tronic beam-splitter and second order interference reveal- ing particle indistinguishability is observed in the detec- tion statistics at the beam-splitter output [22]. Similar ex- periments have been repeated with the leviton source dis- cussed below [23,24]. The 2007 first realization of a sin- gle electron source has triggered many theoretical works [25,26,27,29,30,31,32,33,34] as it opened the possibil- ity to test dynamical quantum transport at the single elec- tron level, investigating single and two-electron coherence, non-local entanglement, waiting time statistics or quantum heat fluctuations. However the mesocopic capacitor source is delicate to operate and to fabricate (tuning two identi- cal energy levels and emission time to demonstrate elec- tron undistinguishability in [22] was a tour de force). Like single photon sources it displays a fundamental quantum uncertainty in emission time, a quantum jitter measured by high frequency shot noise in [35]. A simpler comple- mentary approach to single electron sources for electron quantum optics came more recently with the realization of voltage pulse sources. The voltage pulse source is based on a simple principle. A voltage pulse V (t) is applied on the contact of a quantum conductor where electrons can propagate in few modes. According to ac quantum transport laws, if all other con- tacts are grounded, a current I(t) = e2/hV (t) is injected form the contact in each mode emitted from this contact (disregarding spin). Then, by tuning the amplitude and du- ration of the voltage pulse such that R ∞ −∞I(t)dt = e a sin- gle charge is injected. The procedure seems too simple to think it can give correct results. Indeed, if a single charge is actually injected, it is in general accompanied by unwanted neutral excitations in the form of electron-hole pairs result- ing from the perturbation of the electrons already present in the conductor (the Fermi Sea). However, nearly twenty years ago, in their quest for a clean electron generation model for studying the Full Counting Statistics in quan- tum conductors, L. Levitov and collaborators discovered that, if the voltage pulse has the shape of a Lorentzian, a clean single electron injection (without extra neutral ex- citations) can be realized [36,37,38,39]. The Lorentzian voltage pulse single electron source has been experimen- tally realized in 2013 by the present authors and their team at CEA Paris-Saclay and the predicted minimal excitations states, now called levitons, have been carefully character- ized by their minimal shot noise [23]. The single electron leviton source is presented in more detailed and reviewed", + "source has been experimen- tally realized in 2013 by the present authors and their team at CEA Paris-Saclay and the predicted minimal excitations states, now called levitons, have been carefully character- ized by their minimal shot noise [23]. The single electron leviton source is presented in more detailed and reviewed in the next parts of the present paper. In order to conclude this part I will refer the reader to Fig. 2 which shows a comparison of the Mesoscopic capacitor and Leviton single electron sources, both being suitable to perform electron quantum optics. The figure shows the complementary aspects of the two sources. The mesoscopic capacitor source emits energy resolved elec- D Vexc D Time 2eV (t) exc VG 1 2 3 1 2 3 a) b) Vexc Vexc Vexc 8 16 24 32 0 8 16 24 32 -1 0 1 8 16 24 32 2eVexc / D x3 1 0 time (ns) I (a.u) time (ns) x1 x1.5 time (ns) VG D = 0.03 t = 0.9 ns t = 3.6 ns D = 0.005 D = 0.002 t = 10 ns Figure 1 a) Radio-frequency pulse applied on the top gate (inset of the figure). (1) Starting point: the fermi level lies between two discrete energy levels of the quantum dot. (2) 2eVexc(t) is equal to the level spacing ∆. An electron escapes the dot at a well defined and tunable energy. (3) Vexc(t) is brought back to its initial value, a hole escape at energy below the lead Fermi energy . b) Time domain measurement of the average current (black curves) on one period of the excitation signal (red curves) at 2eVexc(t) = ∆for three values of the transmission D. The expected exponential relaxation with time ¯h/D∆fits well the data (blue curve)(figure adapted from [18]). trons while the leviton source is time-resolved. There is a duality of their properties if we interchange time and energy: Lorentzian energy distribution versus Lorentzian time wavepacket, semi-exponential time wavepacket ver- sus semi exponential energy distribution above the Fermi energy. Being energy-resolved, the mesoscopic capacitor source allows to make nice fundamental tests of the en- ergy relaxation and decoherence of a Landau like quasi- particle above the Fermi sea [40]. For the same reason the electrons injected well above the Fermi sea are more prone to decoherence than the levitons whose energy dis- tribution is as close as possible to the Fermi energy [40]. Due to charging effects complicating electron injection, the mesoscopic capacitor source is limited to single charge injection. The voltage pulse source lacks of exact charge quantization (as the charge is tunable) but it can simul- taneously injects any number of electrons while keeping the levitonic minimal excitation property. Interestingly, it was theoretically shown that if the mesoscopic capacitor is driven adiabatically (i.e. the energy level not suddenly risen above the Fermi energy but slowly risen at constant Copyright line will be provided by the publisher 4 D.C. Glattli et al.: levitons EF ε t mesoscopic capacitor electron source: t e t Γ − ∝ ) ( ψ 2 2", + "driven adiabatically (i.e. the energy level not suddenly risen above the Fermi energy but slowly risen at constant Copyright line will be provided by the publisher 4 D.C. Glattli et al.: levitons EF ε t mesoscopic capacitor electron source: t e t Γ − ∝ ) ( ψ 2 2 0 2 ) ( 1 ) ( Γ + − ∝ ε ε ε ψ t iw t t + ∝1 ) ( ψ )(t V EF w e . 2) ( ε ε ψ − ∝ ε voltage pulse source electron source: leviton Figure 2 Top figure: the mesoscopic capacitor source (energy-resolved source). Its energy distribution is lorentzian and the time wavepacket envelope is semi- exponential. Bottom Figure: the voltage pulse leviton source. Its energy distribution is semi exponential and the time wavepacket is lorentzian. The two sources are suitable for electron quantum optics. They show a duality when in- terchanging time and energy speed), the Lorentzian energy shape of the level gives rise to a Lorentzian wavepacket similar to a leviton with the emitted charge fundamentally fixed to the charge e [27]. 2 Levitons: time-resolved single electron with minimal excitation property. 2.1 principle We consider a single mode conductor, spin disregarded. The following results can be directly ex- tended to multiple modes including spin. The single mode can be an edge channel of the integer quantum Hall ef- fect for filling factor one. It can also be a single elec- tronic mode of a 2D conductor spatially filtered by a nar- row constriction like a Quantum Point Contact inserted in the middle of the 2D conductor. When a time dependent voltage V (t) is applied, say on the left contact of conduc- tor while the opposite contact is grounded, electrons emit- ted at energy ε below the Fermi level of the contact, and experiencing the voltage, acquire a time dependent phase φ(t) = e R t −∞V (t′)dt′/¯h. Because of the time depen- dency, energy is not conserved and the electron are scat- tered in a superposition of quantum states of different pos- sible energies. The amplitude of probability to have the en- ergy displaced from ε to ε + δε is : p(δε) = Z ∞ −∞ e−iφ(t)eiδεt/¯hdt (1) and the probability P(δε) = |p(δε)|2. Eq. 1 is the basis of the Floquet scattering theory developed by M. B¨uttiker and M. Moskalets for periodically driven conductors [28]. For V (t) a voltage pulse, the injected charge q is q = R ∞ −∞I(t)dt. For a spinless single mode, for which I(t) = e2V (t)/h, this gives the Faraday flux (or action): Z ∞ −∞ eV (t)dt = (q/e)h (2) and the phase increment: ∆φ = φ(−∞) −φ(∞) = 2π(q/e) (3) In general p(δε) have non zero values for both positive and negative δε. Consequently all electrons of the left contact Fermi sea are displaced up and down in energy. Compared with the reference Fermi sea of the grounded right con- tact, this gives both electron and hole like excitations. At zero temperature, with sharp Fermi sea distribution, one can", + "positive and negative δε. Consequently all electrons of the left contact Fermi sea are displaced up and down in energy. Compared with the reference Fermi sea of the grounded right con- tact, this gives both electron and hole like excitations. At zero temperature, with sharp Fermi sea distribution, one can quantify the number of electrons Ne and hole Nh ex- citations created by the pulse. They are given by: Ne = Z ∞ 0 δεP(δε)d(δε) (4) Nh = Z 0 −∞ (−δε)P(δε)d(δε) (5) The charge introduced by the pulse in the conductor is q = e(Ne −Nh) and the total number of excitations is Nexc = Ne + Nh. For a single electron source injecting electrons in the form of a minimal state called leviton, it is mandatory to have no hole excitations creation, i.e. Nexc = Ne = 1 such that only excitation is the injected electron, as schemati- cally shown in Fig 3. To do that, the Fourier transform of the phase term e−iφ(t), which in Eq. 1 gives p(δε), must be zero for all negative δε. According to standard complex integration technique, this implies the phase term e−iφ(t), prolongated in the complex plane must have no pole in the lower half plane and at least one pole in the upper half complex plane. The simplest expression for the phase term with only one pole, at say t = iw, is: e−iφ(t) = t + iw t −iw (6) With this form, the phase increment is ∆φ = 2π and, from Eq.3, the associated charge is q = e while the phase deriva- tive is a Lorentzian with 2w full width at mid-height. The corresponding voltage pulse is therefore a Lorentzian: V (t) = ¯h e 2w t2 + w2 (7) Copyright line will be provided by the publisher pss header will be provided by the publisher 5 energy F E ) (t Vac ε ) ( / t i t i x k i e e e φ ε − − h ε ε ε ∆ + ε ∆ EF ) ( ~ ε f ε el. hole energy F E ) (t Vac ε ) ( / t i t i x k i e e e φ ε − − h ε ε ε ∆ + ε ∆ EF ) ( ~ ε f ε el. hole ) (t V ) (t V arbitrary pulse Lorentzian pulse Figure 3 Top figure: effect of an arbitrary pulse on the Fermi sea. Electrons are scattered to different ener- gies above and below their initial energy. Electron hole excitations are created, as schematically represented in the energy distribution ef(ε). Bottom figure: applying a Lorentzian shape voltage pulse generates a minimal exci- tation state called Leviton , no hole excitation is created. Eqs. 6,7 uniquely define the type of voltage pulse needed to generate a single charge leviton. Generalization to multiple electron injection in the form of levitons in- jected at time tk with width wk is possible by introducing extra poles in the upper complex plane, such that e−iφ(t) =", + "created. Eqs. 6,7 uniquely define the type of voltage pulse needed to generate a single charge leviton. Generalization to multiple electron injection in the form of levitons in- jected at time tk with width wk is possible by introducing extra poles in the upper complex plane, such that e−iφ(t) = N Y k=1 t −tk + iwk t −tk −iwk (8) V (t) = N X k=1 ¯h e 2wk (t −tk)2 + w2 k (9) The absence of poles in the lower half part of the complex plane in 8, ensures that the multiple electron injection still forms a minimal excitation state. Simultaneous injection of N electrons in the form of levitons is also possible by sim- ply increasing by N the pulse amplitude. The phase term acquires a single pole of order N, in the upper half com- plex plane e−iφN(t) = ( t+iw t−iw)N. In [41] it was remarked that such phase modulation generates N orthogonal enve- lope wave-functions of the form φk(u) = (u+iw)k−1 (u−iw)k , with u = t −x/vF, k = 1, N and the resulting quantum state generated from the voltage pulsed Fermi sea is a slater de- terminant made of these N wave-functions. The voltage pulse charge injection method also al- lows to inject arbitrarily a non-integer charge q. Keeping a Lorentzian pulse shape, this gives the phase term: e−iφ(t) = \u0012t + iw t −iw \u0013q/e (10) One immediately sees that if q is not integer, the resulting fractional power gives non-analytic properties to the phase term on the real axis. As a consequence, the Fourier trans- form of e−iφ(t) acquires non-zero values for positive and negative δε giving both electron and hole excitations. This situation was named in Levitov’s original paper [36] ”dy- namical orthogonality catastrophe” in reference to the An- derson orthogonality catastrophe problem but in the time domain. To summarize, Lorentzian voltage pulses generate clean minimal excitation states called levitons carrying an integer number of charge only. Other pulse shape intro- duce extra excitations in the form of neutral electron hole pairs. 2.2 Evidence and characterization of levitons 2.2.1 Minimal noise for minimal excitation states In a practical experiment, as in [23], periodic injection is chosen. For an injection frequency ν, period T = 1/ν, the average current is I = eν. The periodicity implies that P(δε) = Σ∞ l=−∞Plδ(δε−lhν), the discrete Pl being the l- photo-absorption probabilities. To experimentally demon- strate that the right pulse shape, the Lorentzian, generates a minimal excitation state, the expected levitons were sent to an artificial quantum impurity, a Quantum Point Con- tact, which partially transmits a single electronic mode. Its mode transmission D was tunable by gates and mea- surable by dc conductance. The idea is to take advan- tage of the partitioning of electron and hole excitations by the QPC into transmitted and reflected modes. This fun- damentally quantum probabilistic effect generates what is known as quantum shot noise,where shot noise means cur- rent noise. It can be shown that the shot noise at zero tem- perature is a direct measure of", + "of electron and hole excitations by the QPC into transmitted and reflected modes. This fun- damentally quantum probabilistic effect generates what is known as quantum shot noise,where shot noise means cur- rent noise. It can be shown that the shot noise at zero tem- perature is a direct measure of the total number of excita- tions Nexc = Ne + Nh [36,39,42]. The spectral density of the current noise SI is given by: SI = 2e2νD(1 −D)(Ne + Nh) (11) By comparing the noise generated with sinewave, square wave and Lorentzian pulses, ref.[42] showed that for integer charge, thermal corrections included, the shot noise of Lorentzian pulses is indeed minimal and cor- responds exactly to the noise of a clean single charge. Thus minimal excitation states, levitons, generate minimal noise. 2.2.2 energy spectroscopy: absence of hole ex- citations Energy spectroscopy is another way to charac- terize the levitonic state. Indeed, contrary to sinewave or square wave pulses, it is expected that Lorentzian pulses carrying charge e generate no hole excitation. It is thus Copyright line will be provided by the publisher 6 D.C. Glattli et al.: levitons P-2 P0 P-1 P1 !\"(#) EF Energy Electrons Holes P0 P-1 P-2 = 0 P1 EF !\"(#) Energy Electrons No hole SINE WAVE PULSE LORENTZIAN PULSE Figure 4 Top figure: shot noise spectroscopy of a sine wave pulse. Since an equal number of electron and hole excitations is created the noise spectroscopy is symmetric with respect to VR = 0.The schematic zero temperature en- ergy distribution ef(ε) is shown on the left. Bottom figure: shot noise spectroscopy of a lorentzian pulse. The num- ber of holes is minimized to zero and the shot noise spec- troscopy is asymmetric. The figures are adapted from [23] important to show the absence of hole excitations for en- ergies below the Fermi energy. Again, the partition shot noise of a Quantum Point Contact can be used to demon- strate this. The idea is to add a dc voltage VR on the op- posite right contact. Under negative bias, the hole emitted from the periodically driven left contact in the energy range −eVR < ε < 0 will no longer contribute to noise and the corresponding shot noise variation will measure their num- ber. For positive VR, electrons emitted in the energy range 0 > ε > VR will anti-bunch with electron excitations com- ing from the driven left contact and no noise is produced. Again, the noise variation with VR will give a measure of the electron excitations. With a finite VR, the shot noise is still given by expression Eq. 11 but for VR > 0, Ne is replaced by Ne,VR = R ∞ EVR δεP(δε)d(δε) and Nh un- changed while for VR < 0, Ne is unchanged by Nh relaced by Nh,VR = R −e|VR| −∞ (−δε)P(δε)d(δε). Fig. 4 shows shot noise spectroscopy comparison of sine wave and Lorenztian pulses and comparison with pre- dictions. For a sine wave, a symmetric noise variation with VR is observed due to symmetric electron and hole contri- bution which strikingly", + "by Nh relaced by Nh,VR = R −e|VR| −∞ (−δε)P(δε)d(δε). Fig. 4 shows shot noise spectroscopy comparison of sine wave and Lorenztian pulses and comparison with pre- dictions. For a sine wave, a symmetric noise variation with VR is observed due to symmetric electron and hole contri- bution which strikingly differs from the asymmetric noise variation for Lorentzian due to the absence of holes. 2.2.3 Time domain characterization: Hong Ou Mandel correlations In the celebrated Hong Ou Man- del photonic experiment, the authors were at first inter- ested to determine the time shape of the photon pairs emitted by parametric down conversion of a Laser pulse sent to a non-linear crystal. To do this, they took advan- tage of the photon bunching effect which is expected to occur when two undistinguishable photons arrive at the same time and mix on a beam-splitter. According to Bose statistics, both photons prefer to exit at the same beam- splitter output and experimentalists repeating the exper- iment will find fluctuations ⟨(∆N)2⟩in the detection photon number N due to the partitioning by the beam- splitter of two bunched photons. This is twice the particle noise expected if the photons were separately arriving and partitioned by the beam-splitter. By varying the delay τ between the photon arrival, one progressively goes from the first to the second situation and the variation is sim- ply given by the overlap of the photon wave-functions ψ which partially mix for finite τ in the beam-splitter, i.e. ⟨(∆N)2⟩τ = ⟨(∆N)2⟩τ=∞(1 + |⟨ψ(0)|ψ(τ)⟩|2). This provided a measure of the photon wave-packet extension and a nice evidence of bosonic quantum statistics. A simi- lar trick can be done with electrons which, being fermions, anti-bunch. When in coincidence (τ = 0) they will always take different outputs to satisfy Pauli exclusion and the particle noise is zero. Hong Ou Mandel (HOM) correlations can be realized using levitons periodically sent with a relative delay τ from opposite contacts toward a Quantum Point Contact playing the role of the electronic beam-splitter while measuring the cross correlation shot noise between output leads. One ex- pects SI = 2e2νD(1 −D)2(1 −|⟨ψ(x)|ψ(x −vF τ)⟩|2) (12) where ψ is the leviton wavefunction. If doubly charge levi- tons are sent, one expects: SI = 2e2νD(1 −D)2(2 −|⟨ψ1(x)|ψ1(x −vF τ)⟩|2− |⟨ψ2(x)|ψ2(x −vF τ)⟩|2) (13) where ψ1,2 are the first two orthogonal levitonic wave functions [41] carrying the two incoming electrons. Re- markably, the voltage pulse electron injection technique allows the generation of an arbitrary number of electrons and for performing N-electrons HOM correlation. In this case, Eq. 13 generalizes easily. Because of constraints due to Fermi statistics, a direct comparison between HOM cor- relations with N-photon Fock states [59] and those with N-electron minimal excitation states cannot be done. This situation opens new perspectives in quantum mechanics. Fig. 5 shows the HOM shot noise experiment done with single and doubly charged levitons. The comparison with expected HOM noise variation is excellent without adjustable parameters. This provides a good check of the Copyright line will be provided by the publisher pss header will be provided", + "in quantum mechanics. Fig. 5 shows the HOM shot noise experiment done with single and doubly charged levitons. The comparison with expected HOM noise variation is excellent without adjustable parameters. This provides a good check of the Copyright line will be provided by the publisher pss header will be provided by the publisher 7 0 ,0 0 0 ,2 5 0 ,5 0 0 ,7 5 1 ,0 0 0 ,0 0 ,2 0 ,4 0 ,6 0 ,8 1 ,0 1 ,2 1 ,4 1 ,6 T im e d e la y (p e rio d u n it) τ/T S I coll. /S I 0 L o re n tz ia n W a v e α = 2 ( 2 e - ) L o re n tz ia n W a v e α = 1 ( 1 e - ) τ τ SI HOM/SI 0 (1 e ) (2 e ) Figure 5 Top figure: HOM experiment for one and two electrons injected in the separate input of a beam-splitter. The HOM shot noise resulting from partitioning of anti- bunched electrons is plotted versus the arrival time delay τ giving access to the wavefunction overlaps. Note the ex- cellent agreement with the expected theory (solid line). Bottom figure: HOM experiment with two different pulses. The sine pulse creating a large amount of electron hole ex- citations, the HOM shot noise is enhanced. The figures are adapted from [42]. levitonic states with one or two electrons in the time do- main. The HOM shot noise of colliding sine-wave pulses is also shown for comparison. Here, because sine-wave pulses give rise to charge pulses accompanied with a cloud of electron-hole pairs and all these excitations interfere in the electron beam-splitter, the physical information to ex- tract is not clear (except to confirm the good agreement with HOM shot noise theoretical modeling [42]). The leviton HOM correlations give remarkably 100% noise suppression at τ = 0. This contrasts with the small HOM dip observed in ref [22] where electron are injected from a mesocopic capacitor. In the latter case, the reason for a weak HOM noise suppression is not due to the dif- ferent nature of the source but is due to the propagation in the leads between the source and the beam-splitter. In [22] the leads are made of two chiral edge channels (In- teger Quantum Hall regime at filling factor 2) and elec- trons are injected in the outer edge channel. A (pseudo)- spin charge separation occurs where the incoming elec- tron fractionalizes. This induces decoherence and energy relaxation which spoils the otherwise naively expected full HOM noise suppression. Working with a single edge or LORENTZIAN SINE-WAVE Figure 6 Temperature dependence of the HOM shot noise for lorentzian (left) and sinewave (right) pulses. By plotting the ratio of the HOM noise to the theoretical zero temper- ature HOM noise versus the HOM delay τ, one observes that, contrary to the sinewave pulses, for levitons the HOM noise shape versus τ is not affected by thermal excitations. Figures", + "lorentzian (left) and sinewave (right) pulses. By plotting the ratio of the HOM noise to the theoretical zero temper- ature HOM noise versus the HOM delay τ, one observes that, contrary to the sinewave pulses, for levitons the HOM noise shape versus τ is not affected by thermal excitations. Figures adapted from [41] putting the source closer to the beam-splitter would pro- duce a better HOM shot noise dip. In [42], it was remarked that the HOM noise shape versus τ does not depends on temperature. This was ac- tually experimentally observed [41]. This remarkable de- coupling occurs only for single charge levitons but not for sine wave pulses nor for doubly charged levitons. The ro- bustness of the 100% HOM dip to temperature for any voltage pulse shape and of the HOM leviton noise curve versus τ to temperature effects and decoherence has been recently theoretically studied in [60]. Fig 6 shows exper- imental HOM noise measurements demonstrating the de- coupling between the temperature and the delay τ for the case of a leviton and shows the absence of decoupling for sinewave pulses. 2.2.4 Full characterization: Wigner function of a leviton In the last two sections levitons have been charac- terized in time or in energy. But a wave function is a more complex object. Because of the fundamental quantum un- certainty of conjugated variables ( energy versus time or momentum versus position), standard quantum measure- ments gives in general information on the wavefunction versus one variable only. Having all information on the wave function requires a lot of different measurements to ’see’ the wavefunction under all aspects. This is done using Quantum State Tomography (QST). It consists in measur- Copyright line will be provided by the publisher 8 D.C. Glattli et al.: levitons 0 0.1 0.2 0.3 energy ( hν0 ) −2 −1 0 1 2 3 4 5 k = 2 Exp. Theory −0.1 0 0.1 delay τ / T 0 0.5 1 k=1 η=0.095 −0.04 −0.02 0 0.02 0.04 delay τ / T 0 0.5 1 k=2 η=0.0425 0 0.02 0.04 0.06 eVR / hv0 −1 0 1 2 3 k = 1 exp. fit theory 10-30A2/Hz Exp. Theory Figure 7 The off diagonal terms of the energy density matrix of a leviton (bottom figures) are extracted from the shot noise (middle figures) measured when performing the HOM interference of a leviton mixing (colliding) in a beam-splitter with a small sine-wave pulse at frequency ν (left figure) and 2ν (right figure). The top figures show the noise variation with the time delay τ between the small sinewave pulse and the periodic levitons. The oscillations, and the doubling of the period at frequency 2ν with respect to ν, prove the existence of non-diagonal coherence terms in the energy density matrix. The figures are adapted from [24]. ing the energy density matrix ̺(ε′, ε) =< ψ†(ε′)|ψ(ε) > or the coherence < ψ†(t′)|ψ(t) > to which the pure Fermi sea contribution has been subtracted. From this, one can reconstruct the Wigner function W(t, ε) which represents a quasiprobability distribution of the", + "matrix. The figures are adapted from [24]. ing the energy density matrix ̺(ε′, ε) =< ψ†(ε′)|ψ(ε) > or the coherence < ψ†(t′)|ψ(t) > to which the pure Fermi sea contribution has been subtracted. From this, one can reconstruct the Wigner function W(t, ε) which represents a quasiprobability distribution of the state versus the two- dimensional phase space of conjugated variables (ε, t) ( or (p, x)) . W(t, ε) = Z ∞ −∞ dδ̺(ε + δ/2, ε −δ/2)e(−iδt/¯h) (14) For periodic injection, only the non-diagonal elements of ρ differing by a multiple of the fundamental frequency ε′ −ε = lhν are non zero. This implies that the Wigner function is a periodic function of time and this conve- niently restricts the number of measurements to be done. A Quantum State Tomography procedure for Measurements of ̺(ε′, ε) has been theoretically proposed by C. Grenier in [44] and has been experimentally implemented and re- ported by the present authors in [24]. First let us remark that the Shot noise spectroscopy makes a measurement of the diagonal term of the energy density matrix as one can deduce from the measurements the electron distribution function ef(ε) = ̺(ε, ε). In order to access the off-diagonal elements where the energies differ by khν, the same shot noise spectroscopy technique is used but, following [44], instead of only applying a dc voltage VR on the right reser- voir, a small amplitude Vac sine-wave voltage is added at frequency kν. Electrons emitted by the right reservoir at energy ε are thus is weak amplitude superposition of states of energies ε ± khν. Mixed in the beam-splitter, they anti- bunch with the ε and ε ± khν leviton energy components. Finally, Introducing a delay τ between the small ac sine- wave signal and the leviton pulses, like in the HOM ex- periment, provides a way to give a convincing signature of the interference of the ε and ε + khν terms in the result- ing shot noise variation. In [24] only the first two harmonic terms were measured (k = 0, 1, 2) but the third one could have been measured as well. The expected shot noise dif- ference ∆SI obtained when the noise is measured with Vac on and off is given by: ∆SI =S0 I 2kη cos(2πkντ). . Z eVR 0 (̺(ε, ε + khν) −̺(ε, ε −khν))dε/hν (15) from which the off diagonal terms can be extracted. Fig 7 shows the noise variations for k = 1, 2. The characteris- tic doubling of the period with τ ensures that interference terms are indeed measured. This demonstrates the fully co- herent nature of the levitonic quantum state. The deduced off-diagonal components (bottom of the figure) are then used to extract from Eq. 14 the leviton Wigner function truncated to the first two harmonic temporal components as shown in Fig. 8 3 Perspectives Levitons seem to be the ideal source for electron quantum optics and offers wide perspectives. The excellent ability to synchronize leviton sources could be exploited to realize leviton flying qubits using the edge state topology.", + "truncated to the first two harmonic temporal components as shown in Fig. 8 3 Perspectives Levitons seem to be the ideal source for electron quantum optics and offers wide perspectives. The excellent ability to synchronize leviton sources could be exploited to realize leviton flying qubits using the edge state topology. As an example, we propose the follow- ing CNOT quantum gate coupling two flying qubits (a) and (b). Starting with the sate |1a⟩|1b⟩by injecting levi- tons on the appropriate contacts, the output is expected to be |1a⟩|1b⟩+ |0a⟩|0b⟩a Bell state. Here we have used a coulomb coupling which provides a conditional π phase shift when an electron on the lower edge of qubit (a) per- forms a distant scattering with an electron propagating on the upper edge of qubit (b). Fig. 9 schematically displays such a quantum circuit with chiral ballistic electrons to per- form the CNOT quantum gate. Leviton single electron sources have stimulated many theoretical works. The time domain control and the special minimal excitation state property leads to well posed prob- lems not accessible to dc voltage electron sources. Here we will cite only a few examples of many theoretical pre- dictions concerning levitons. For example, the non-local Copyright line will be provided by the publisher pss header will be provided by the publisher 9 WDF experimental Wigner function Experiment Theory Figure 8 Experimental Wigner function extracted from the off-diagonal components of the density matrix (adapted from [24]). entanglement of electrons emitted by two independent dc voltage sources, as first considered in [45] in a two particle Hanbury Brown Twiss set-up, was difficult to characterize and found very fragile with respect to thermal fluctuations [46]. The same problem has been recently recently revis- ited using leviton sources and it has been shown that non local characterization of entanglement is possible without the need of post-selection [61]. Another example is the generation of neutral levitons pairs by a controlled time variation of the transmission amplitude of a punctual scat- ter like a QPC which has have been first predicted in [48]. The principle is based on a controlled variation of the QPC transmission d(t) and reflection r(t) amplitudes. Writing the scattering amplitudes in the form of d(t) = sin φ(t) and r(t) = i cos(φ(t), to ensures unitarity of the QPC scattering matrix, and choosing a phase variation φ(t) cor- responding to that used to produce levitons with voltage pulses, see Eq. 6, neutral leviton anti-leviton pairs are pre- dicted to be emitted in the adjacent leads from the mod- ulated scatter. These pairs are expected to be entangled [49]. Embedding, these neutral leviton pairs in a Mach- Zehnder electronic interferometer, their entanglement has been theoretically demonstrated and the entanglement de- tection theoretically shown possible in [50]. Levitons can also be useful for new quantum detec- tion schemes. For example, the Full Counting Statistics (FCS) of levitons partitioned by a quantum conductor has been theoretically shown detectable by sending synchro- nized levitons in a coupled Mach-Zehnder interferometer [61]. Here, remarkably,no noise measurements are needed, )(t V )(t V t t", + "be useful for new quantum detec- tion schemes. For example, the Full Counting Statistics (FCS) of levitons partitioned by a quantum conductor has been theoretically shown detectable by sending synchro- nized levitons in a coupled Mach-Zehnder interferometer [61]. Here, remarkably,no noise measurements are needed, )(t V )(t V t t rail 0a rail 1a rail 1b rail 0b conditional phase π rail 0a rail 1a rail 1b rail 0b (1a) (0a) (0b) (1b) CNOT operation with Levitons Figure 9 Schematic representation of a CNOT quantum gate with Levitons. Two Levitons are injected in rail 1a and 1b. The output state is a maximally entangled state. The vertical dashed line schematically indicates the Coulomb interaction which could provide a conditional π phase shift in the two particle (two-qubit) wavefunction. only dc current measurements are necessary to access the electron FCS. Regarding interaction, it has been predicted that levi- tons can also be generated in an interacting 1D electron system [39] called a Luttinger Liquid (LL). A nice exam- ple is given by the chiral edge states of the Fractional quan- tum Hall effect (FQHE), whose best understood state, the Laughlin state, occurs at filling factor 1/3 [51]. Here the current is carried by a single edge channel. This is a chi- ral Luttinger liquid which is expected to generate elemen- tary excitations carrying fractional charge e∗= e/3 [52] and showing anyonic statistics [53] in full correspondence with the bulk Laughlin excitations. This regime has been considered in [54] and more recently in [55]. Counterin- tuitively and contrary to what was suggested in [39], no fractionally charged levitons can be generated in a frac- tional edge but only integer charge leviton. Indeed, it is mandatory that the total phase variation resulting from the voltage pulse be not a fraction of 2π in order to have only electron like excitations with no holes and thus a mini- mal excitation property. This implies that the Faraday flux, which in the 1/3 FQHE regime writes R e ∗V (t)dt be equal to nh. As I(t) = e ∗(e/h)V (t) the total injected charge for 2nπ phase shift is thus q = ne, where n an in- teger. Can these integer charge minimal excitation states be used to probe the fascinating fractionally charge ex- citations? Having the time domain control of e∗excita- Copyright line will be provided by the publisher 10 D.C. Glattli et al.: levitons tions could open new way to probe their anyonic statis- tics via Hong Ou Mandel correlations or other two-particle interferometry protocol. In Fig. 10 we propose a possible source of on-demand anyons: integer charge levitons are first generated and then sent to a QPC which is tuned in the weak backscattering regime (i.e. shows a weak reflec- tion probability). In this regime, shot noise measurements, using dc voltage source to create an incoming flux of in- teger charges, have shown that a Poissonian emission of fractional charges e∗can be emitted to contribute to the backscattering current [56],[57]. We conjecture that, sim- ilarly, an integer charge leviton would be partitioned into an e/3", + "shot noise measurements, using dc voltage source to create an incoming flux of in- teger charges, have shown that a Poissonian emission of fractional charges e∗can be emitted to contribute to the backscattering current [56],[57]. We conjecture that, sim- ilarly, an integer charge leviton would be partitioned into an e/3 backscattered charge and a 2e/3 transmitted charge. We think this situation would lead to the generation of levi- tonic like charge pulses carrying fractional charge. Here, the deterministic emission of fractionally charge levitons would be lost because of the Poissonian backscattering statistics but the time resolution and the time domain con- trol inherited from the initial integer charge levitons will be kept and could be used for further experiments like Hong Ou Mandel collision of anyons to probe their quan- tum statistics. In another perspective it is interesting to address the physics of non-integer charge pulses. Only voltage pulses can realize this situation while quantum dot based charge sources can only emit integer charge particles. Indeed by continuously tuning the voltage pulse amplitude on a con- tact one can generate any value of the injected charge. It was shown from Eq. 10 that only an integer q gives rise to a minimal excitation state. Injecting fractional charges, even using Lorentzian pulses can not be viewed as pure levitons. However non-integer charge pulses created by Lorentzian voltage pulses have interesting properties which deserve attention. In [58] it was shown that half-levitons, i.e. charge e/2 created by Lorentzian voltage pulses, minimize the shot noise of a Superconducting/Normal conductor junction the same way as integer charge levitons minimize the noise in a purely normal conductor. Here this occurs in the en- ergy sub-gap regime where, while the normal current is suppressed by the superconducting gap, a small current is allowed via the so-called Andre’ev reflection mecha- nism. In Ref. [60], the nature of half integer Lorentzian charge pulses have been studied and it has been shown that they can form remarkable fractionally charged zero- energy single-particle excitations states. Separating the half-leviton into a e/2 charge part and its accompanying neutral electron-hole cloud, [60] showed that a half-leviton and a anti-half leviton mixed in a semi-transparent elec- tronic beam-splitter can elastically annihilate, a property not shared with ordinary distinguishable electron and hole excitations. Non-integer charge pulses behave differently than in- teger pulses in electronic Mach-Zehnder (MZI) or Fabry- Prot (FPI) interferometers. This was first noticed in the work of [62] where, from numerical dynamical simula- tions, it was remarked that, for pulses having an extension smaller than the MZI arm difference or the FPI perime- ter, the transmission was dependent on the charge. For ex- ample, the Fabry-Prot transmission shows oscillations with the charge q modulo e. The explanation was found in the phase difference 2π(q/e), see Eq. 3, between the front and the back of the charge wave-packet which combines with the orbital phase accumulated in the interferometer. Strik- ing effect on the interference visibility of the current in an electronic Mach-Zehnder interferometer have been also found in [63]. For example, while the visibility as", + "see Eq. 3, between the front and the back of the charge wave-packet which combines with the orbital phase accumulated in the interferometer. Strik- ing effect on the interference visibility of the current in an electronic Mach-Zehnder interferometer have been also found in [63]. For example, while the visibility as a func- tion of an external magnetic flux vanishes when the injec- tion period corresponds to have exactly one electron in the MZI for charge q = e, Ref. [63] found that for q = e/3 the visibility vanishes when there are exactly three fractional charge pulses in the MZI. More generally robust visibility cancelation occurs when there are p charges for fractional charges q = ek/p. Similar results are found in a FPI in- terferometer. This intriguing properties call for looking at possible anyonic or parafermionic quantum statistics that may obey these charge pulses. The above mentioned perspectives are only a few di- rection and do not pretend to give a fair exhaustive account of the impressive literature that levitons have triggered and we apologize for missing references and topics. We think that we are only walking on the surface of a Fermi Sea Ice- berg. Acknowledgements Support from the ERC Advanced Grant 228273 MeQuaNo is acknowledged. References [1] L. J. Geerligs, V. F. Anderegg, P. A. M. Holweg, J. E. Mooij, H. Pothier, D. Esteve, C. Urbina, and M. H. Devoret, Phys. Rev. Lett. 64 , 2691 (1990) [2] H. Pothier, P. Lafarge, P.F. Orfila, C. Urbina, D. Esteve, and M.H. Devoret, Physica B: Condensed Matter 169 , 573 (1991) [3] Mark W Keller, John M Martinis, Neil M Zimmerman, and Andrew H Steinbach, Applied Physics Letters 69 (1996). [4] Jukka P. Pekola, Olli-Pentti Saira, Ville F. Maisi, Antti Kemppinen, Mikko Mttnen, Yuri a. Pashkin, and Dmitri V. Averin, Rev. of Modern Phy. 85 , 1421 (2013) [5] L.P. Kouwenhoven, A.T. Johnson, N.C. van der Vaart, C.J.P.M. Harmans and C.T. Foxon, Phys. Rev. Lett., 67, 1626 (1991) [6] M. D. Blumenthal, B. Kaestner, L. Li, S. Giblin, T. J. B. M. Janssen, M. Pepper, D. Anderson, G. Jones, and D. A. Ritchie Nat Phys 3 343 (2007) [7] Akira Fujiwara, Katsuhiko Nishiguchi, and Yukinori Ono Appl. Phys. Lett. 92 042102 (2008) [8] S.?P. Giblin, M. Kataoka, J.?D. Fletcher, P. See, T.?J.?B.?M. 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B 56, 15180 (1997) Copyright line will be provided by the publisher pss header will be provided by the publisher 11 transmitted backscattered incoming QPC (WB) e/3 leviton (e) i n c o m anyon (e/3) (2e/3) V(t) ν=1/3FQHE state ν=1/3FQHE state Lorentzian pulse Figure 10 Proposal for a time-resolved anyonic source in the", + "Copyright line will be provided by the publisher pss header will be provided by the publisher 11 transmitted backscattered incoming QPC (WB) e/3 leviton (e) i n c o m anyon (e/3) (2e/3) V(t) ν=1/3FQHE state ν=1/3FQHE state Lorentzian pulse Figure 10 Proposal for a time-resolved anyonic source in the Fractional Quantum Hall regime (filling factor 1/3). Integer charge levitons are first created with a charge e Lorentzian voltage pulse applied on the contact of a 1/3 fractional edge channel. Then, the leviton arrives at a Quantum Point Contact tuned in the weak backscatter- ing (WB) regime (weak electronic reflection probability). 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B 90, 235416 (2014) Copyright line will be provided by the publisher", + "Entanglements of electrons and cavity-photons in the strong coupling regime Ofer Kfir University of Göttingen, IV. Physical Institute, Göttingen 37077, Germany ofer.kfir@phys.uni-goettingen.de Abstract: This work sets a road-map towards an experimental realization of strong coupling between free- electrons and photons, and analytically explores entanglement phenomena that emerge in this regime. The proposed model unifies the strong-coupling predictions with known electron-photon interactions. Additionally, this work predicts a non-Columbic entanglement between freely propagating electrons. Since strong-coupling can map entanglements between photon pairs onto photon-electron pairs, it may harness electron beams for quantum communication, thus far exclusive to photons. Entanglement between the states of propagating quantum entities, and its violation of local realism [1–3], acquires significance technological importance for quantum technologies, such as quantum computation and quantum key distribution [4]. Photon pairs’ polarization, expressed as Bell-states, allow for the transport of quantum entanglement over ever-growing distances, under the sea [5] and in the outer space [6]. Matter beams may offer an alternative carrier of quantum information [7,8], which utilizes a richer set of properties, such as shorter wavelength, or the controllability with electric and magnetic fields. Specifically, Fundamental particles, such as free electrons, are especially relevant for entanglement transport. Unlike atoms and other composite matter, free electrons are decoupled from free-space radiation, and they lack an internal structure to decohere into. However, the quantum control of fundamental-particle beams is at its infancy. Two important phenomena address the energy-exchange of light and electron beams: PINEM and EELS. In PINEM (photo-induced near-field electron microscopy [9]), a strong (and hence classical) laser field accelerates or decelerates electrons in a beam. This concept allows for optical control over the electron quantum wavefunction [10–13], culminating in the prediction [10] and demonstration [14–16] of attosecond-scale electron pulses. Similar effects, such as electron-energy gain spectroscopy (EEGS) [17,18], and effects of the light’s ponderomotive-energy, such as the Kapitza-Dirac effect [19] and electron-phase retarders [20] are also well described by classical laser fields. In the abovementioned effects, adding or removing a photon from the laser field makes no difference, since the electron-photon coupling is extremely weak. arXiv:1902.07209v2 [quant-ph] 29 May 2019 In EELS (electron-energy loss spectroscopy) the coupling can be increased by using metallic nano- structures, where the large polarizability and plasmonic resonances [21] allows for detectable signals while the nanometric features compensate for the momentum mismatch between electrons and the optical excitations (i.e. plasmons, see detailed reviews by García de Abajo [22], by Talebi [23], and references within). Such systems were investigated using rings [24], spheres [22], cubes [25,26] and rods [27,28] geometries, as well as for stacked particles [29], ordered or disordered structures [30–32], and also symmetric [33,34] and symmetry-broken systems exhibiting non-Hermitian phenomena [35]. However, rapid decoherence eliminates entanglement features [36] between the electrons and any excited plasmons, stemming from radiative damping [37], intrinsic dissipation [38] absorption and sensitivity to defects. Alternatively, transparent dielectrics have no theoretical bound for the excitation probability [30]. Optical excitations in such dielectrics can be readily injected, collected or manipulated, but unfortunately, their coupling to electron beams is weak. This article proposes that the phase-matching of", + "damping [37], intrinsic dissipation [38] absorption and sensitivity to defects. Alternatively, transparent dielectrics have no theoretical bound for the excitation probability [30]. Optical excitations in such dielectrics can be readily injected, collected or manipulated, but unfortunately, their coupling to electron beams is weak. This article proposes that the phase-matching of swift electrons to photons confined in a fiber-based dielectric cavity can increase the interaction towards the strong coupling regime, and theoretically investigates emergent entanglement phenomena. The phase-matching bandwidth can isolate even a single cavity mode, allowing for a single channel of energy exchange between matter and radiation. An analytical entanglement model is developed for the electron interaction with a cavity mode, which can apply for any coupling regime, weak and strong. To exemplify novel phenomena arising in the strong coupling regime, this work focuses on two scenarios: first, the entanglement between cavity photons and a traversing electron is investigated and compared rigorously to the known phenomena of PINEM and EELS in the limit of weak coupling. Second, the cavity capability to mediate non-Coulombic entanglement between two distant electrons within a beam is explored. Finally, the strength and spectral properties of the coupling are evaluated quantitatively, for the case of a stadium-like whispering-gallery mode cavity based on a single- mode fiber. The proposed experimental concept for reaching strong coupling is shown in Figure 1 (more details are discussed later on and in Figure 4). The optical excitations are whispering gallery modes (WGM) of a stadium-geometry cavity based on a thin waveguide. An electron beam passing parallel to a straight section of the cavity, which is a single mode fiber, excites a bound mode via the interaction with the mode’s evanescent tails in the vacuum. The conservation of both energy and momentum between the electron and light is fulfilled only for a specific photon energy, for which the electron travels at the mode’s phase velocity. The dispersion in the fiber narrows the interaction’s spectral bandwidth. In the example shown on Figure 1b, a silicon-nitride (Si3N4) cavity is optimized to couple electron beams accelerated to 200 keV with photons having vacuum wavelength near 1.064 µm (ℏ\u0002\u0003 = 1.1654 ). The coupling bandwidth narrows down to only 0.011 eV after a propagation length of 100 µm. Importantly, the momentum-matching condition, also referred to as phase-matching, allows for the signal within the interaction bandwidth to grow coherently, and to approach strong coupling. Figure 1 – Proposed experiment for a narrow-bandwidth strong coupling. (a) Evanescent optical field couples a cavity mode to an adjacent electron. (b) Phase matching between the electron and the cavity-photon limits the coupling to a narrow spectral band. For example, a 100 µm propagation near a Si3N4 cavity limits the coupling bandwidth (expressed as EELS bandwidth) to 11 meV around ℏ\u0002\u0003 = 1.1654 ( = 1064 \u000f\u0010). Further details are in the text. Figure 2 – Electron-photon entanglements patterns. (a) Colormap of \u0011\u0012\u0013,\u0015\u0011 \u0016, the co-incident probabilities of photons with electron-energy gain \u0017ℏ\u0002\u0003, after a strong interaction (\u0019 = 3) with an empty cavity. (b) Rich entanglement features for an initial coherent state |", + "= 1064 \u000f\u0010). Further details are in the text. Figure 2 – Electron-photon entanglements patterns. (a) Colormap of \u0011\u0012\u0013,\u0015\u0011 \u0016, the co-incident probabilities of photons with electron-energy gain \u0017ℏ\u0002\u0003, after a strong interaction (\u0019 = 3) with an empty cavity. (b) Rich entanglement features for an initial coherent state | = 3⟩, in the cavity. Oscillations in the electron spectra co- incident with Fock-states, c.f. inset for \u000f = 2 and \u000f = 3, are absent from the integrated electron spectrum (top axis, red bars) since Fock states are orthogonal. (c) The known electron spectrum for PINEM, |\u0012\u0015|\u0016 = | \u0015(2|!|)|\u0016 (top, red bars), emerges for weak coupling and highly-populated cavity (\u0019 = 0.25, | ⟩= |10⟩), with |!| = |\u0019 |. In this limit the electrons and photon-states separable. Coincident probability, \u0011\u0012\u0013,\u0015\u0011 \u0016, for \u000f photons and electron-energy gain of \u0017ℏ\u0002\u0003, for an electron ultra-strongly coupled (\u0019 = 3) to an empty cavity. (b) same as (a), but with a coherent-state in the cavity, | = 3⟩, results in rich entanglement patterns. The inset shows electron spectral oscillations for co-incident detection with Fock- states \u000f = 2 and \u000f = 3. The overall electron spectrum (top axis, red bars) is smooth since electron spectra co- incident with different \u000f add incoherently. (c) The known electron spectrum for PINEM, |\u0012\u0015|\u0016 = | \u0015(2|!|)|\u0016 (top, red bars), emerges for weak coupling and highly-populated cavity (\u0019 = 0.25, | ⟩= |10⟩), with |!| = |\u0019 |. In this regime the electrons and cavity photons are nearly independent, and hence separable. The analytical model relays on the narrow spectral response of the proposed system. Only photons having angular frequency \u0002\u0003 are phase-matched with the relativistic electron beam. The narrow-bandwidth suppresses finite electron wavepacket [39,40] effects. The quantum state of these two systems can be described as energy-ladder systems with ℏ\u0002\u0003 spacing between the levels: The photon Fock-states in the cavity are represented by |\u000f⟩, a semi-infinite ladder with \u000f ≥0. The electron states, |#\u0015⟩, with #\u0015 = #\u0003 + \u0017ℏ\u0002\u0003, represent gain (\u0017 > 0) and loss (\u0017 < 0) with respect to the “zero-loss energy”, #\u0003. ℏ is the reduced Plank’s constant. Thus, a general state of such electron-photon system can be written as |'⟩= ( ( \u0012\u0013,\u0015|#\u0015, \u000f⟩ ) \u0015*+) ) \u0013*\u0003 . (1) The relation between the state of the system before and after the interaction can be described by the scattering matrix, ,-, as \u0011'./\u0013012 = ,-|'/\u0013/3/01⟩. Neglecting electron dispersion effects allows to write ,- as exchanging energy between the electrons and the photons, ,- = 4567\u00198 = 9:70;<+9∗:7<0;. (2) Here, \u0019 is the coupling strength, >;, >;? are the non-commuting photon ladder operators, and 67, 67? are the commuting electron-energy ladder operators. The commutation, @67, 67?A = 0, result in an algebra similar to scalars, so ,- behaves as a the displacement operator 4(\u0019) [41]. A comprehensive treatment of 67, 67?, and electron dispersion effects can be found in Sections S.4 and S.1 of the supplementary, respectively. The interaction of a relativistic electron with an empty cavity is an important", + "an algebra similar to scalars, so ,- behaves as a the displacement operator 4(\u0019) [41]. A comprehensive treatment of 67, 67?, and electron dispersion effects can be found in Sections S.4 and S.1 of the supplementary, respectively. The interaction of a relativistic electron with an empty cavity is an important and instructive case to consider (see Figure 2a). The state of the combined electron-photon system before any interaction can be written as a pure state, with the electron at the zero-loss energy and no photons. |'/⟩= |#\u0003, 0⟩. (3) As the interaction is a displacement operator, the final state after the first interaction, \u0011'.2, is a coherent state [42], as for plasmons [43], in which energy conservation entangles each optical Fock-state to an equal electron-energy loss, #\u0015 = #+\u0013. \u0011'.2 = ( +|9|D \u0016 \u0019\u0013 √\u000f! |#+\u0013, \u000f⟩ ) \u0013G\u0003 . (4) One can consider eq. (4) as the multi-level electron-photon equivalent of a Bell-pair, \u0011'.2 = (\u0012\u0003|#\u0003, 0⟩+ \u0012H|#+H, 1⟩+ ⋯). Thus, coincidence measurements are expected to expose correlations between the measured electron-energy loss and photon detection. For comparison with EELS experiments, one should consider a weak coupling, where only one energy-loss channel is detectable, with population probability of |\u0019|\u0016. Higher EELS orders [43] necessitates a strong coupling |\u0019|~ 1. In the supplementary, S.2, strong coupling EELS is derived as “field-less PINEM” to touch upon their equivalence. A general feature of the electron-energy distribution is that the average loss is #\u0003 −⟨#⟩= |\u0019|\u0016, in either weak or strong coupling. The one-to-one entanglement between the two entities is expressed as the purely diagonal electron-photon energy-state population in Figure 2a. For a quantum-optics description of PINEM experiments one needs to consider a coherent state | ⟩ with an average number of | |\u0016 photons in the cavity. In this case, the exact final electron-photon quantum state is characterized by eq. (1) with the coefficients \u0012\u0013,\u0015 MNOPQ = 〈#\u0015, \u000f\u00114567\u00198\u0011#\u0003, 〉, \u0012\u0013,\u0015 MNOPQ = |9|D+|T|D \u0016 (\u0013U\u0015)\u0019\u0015 √\u000f! ( V(\u000f + \u0017 + ℓ)! (\u000f + \u0017)! X (−|\u0019|\u0016)ℓ (\u0017 + ℓ)! ℓ! ) ℓ*\u0003 . (5) The supplementary section S.2 details the algebraic derivation. Figure 2b presents the electron-photon spectral probability map for the case of a strong coupling (\u0019 = 3) to a cavity populated with 9 photons in average, = 3. The entanglement correlates diagonally as for an empty cavity, but also includes rich patterns. Specifically, the electron spectrum depends strongly on the coincident Fock state (see inset for \u000f = 2 and \u000f = 3). The photon-averaged electron spectra is smooth (red bars, top axis) , similar to a spatial scattering of atoms off a coherent photon state [44,45]. Consistently with EELS and with the case of an empty cavity, the mean energy-loss is |\u0019|\u0016. The electron energy distribute nearly symmetrically around the mean, with a spectral widths of 4|\u0019 |. This generalizes PINEM [9,46,47], known for its 4|!| bandwidth, symmetric around #\u0003 (special case of |\u0019|\u0016 →0) [10]. ! is the PINEM Rabi-parameter. To exactly retrieve known PINEM spectra, with probability amplitudes \u0012\u0015 = \u0015(2|!|) [10,46], one needs to", + "symmetrically around the mean, with a spectral widths of 4|\u0019 |. This generalizes PINEM [9,46,47], known for its 4|!| bandwidth, symmetric around #\u0003 (special case of |\u0019|\u0016 →0) [10]. ! is the PINEM Rabi-parameter. To exactly retrieve known PINEM spectra, with probability amplitudes \u0012\u0015 = \u0015(2|!|) [10,46], one needs to consider strong optical fields | | ≫1, weakly coupled to the electron beam, \u0019 ≪1 (see detailed calculation in the supplementary section S.2.2.1). \u0015 is the Bessel function of the first kind. Eq. (5) reduces to Bessel-function amplitudes when approximating the square brackets within it as (\u0013U\u0015Uℓ)! (\u0013U\u0015)! ≈ (\u000f + \u0017)ℓ. Thus, one can write (\u000f + \u0017)ℓ(−|\u0019|\u0016)ℓ= ]−^\u0019_(\u000f + \u0017)^ \u0016 ` ℓ . The summation then transforms to a_(\u000f + \u0017)b +\u0015 /\u0015 cdef ⋅ \u0015(2|!|), with the definition. ! = \u0019_(\u000f + \u0017) ≈\u0019| |. (6) Neglecting quantum fluctuations added to the light by the interaction, a_(\u000f + \u0017)b +\u0015 = a_⟨\u000f + \u0017⟩b +\u0015 , decouples the electron and photon states. Thus, the decoupled state |'MNOPQ⟩= | ⟩⊗ \u0015(2|!|)|#\u0015⟩, (7) fully retrieves the known PINEM Bessel-amplitudes decoupled from the driving laser field. This decoupling allows to consider the electron wavefunction, and its laser-modulation. Furthermore, the Rabi-parameter, ! emerges naturally as the product of the coherent state amplitude and the coupling strength. Figure 2c shows the electron-photon spectra for = 10, \u0019 = 0.25, that is, a cavity with 100 photons in average, weakly coupled to the electron beam. The electron spectrum is nearly independent of the photon state, yielding the electron spectral oscillations typical for Bessel amplitudes (see zoomed electron spectrum). The second strong-coupling phenomena exemplified here is non-Coulombic entanglement of two consecutive electrons in a beam, mediated by long lived cavity photons. A lifetime of 10 picoseconds allows excitations of the first electron to affect the second, while suppressing Coulombic interactions between them. The passage of the first electron generates a state as in eq. (4). A second electron with an equal zero- loss energy (marked here ℰ\u0003 to distinct from the first electron) will result in a three-particle state, \u0011'. j+j2 ∑ ∑ \u0012\u0013,\u0015 j+j|#+\u0013, ℰ\u0015, \u000f −\u0017⟩ \u0013 \u0015*+) ) \u0013*\u0003 , characterized by 2-indices \u0012\u0013,\u0015 j+j = 〈#+\u0013, ℰ\u0015, \u0010\u00114567\u00198\u0011#+\u0013, ℰ\u0003, \u000f〉. \u0017 is the energy quanta gained by the second electron and \u000f is the Fock- state index prior to the arrival of the second electron, which is also the final energy state of the first electron |#+\u0013⟩. The final Fock-state of the cavity is |\u000f −\u0017⟩. Thus, \u0012\u0013,\u0015 j+j = \u0019\u0013\u0019\u0015 \u000f! _(\u000f −\u0017)! ((\u000f + ℓ)! (−|\u0019|\u0016)ℓ (\u0017 + ℓ)! ℓ! ) ℓ*\u0003 . (8) Explicit derivation are in the supplementary section S.3. One can think of such an event as PINEM, pumped by the first electron, as apparent in the similarity of eqs. (5) and (8). Figure 3 shows the resulting entanglement features for strong couplings of \u0019 = 1 and \u0019 = 3. The single particle spectra (right axis, red bars) is smooth, while oscillations appear in co-incidence measurements. See the inset of Figure 3b for a spectrum", + "the similarity of eqs. (5) and (8). Figure 3 shows the resulting entanglement features for strong couplings of \u0019 = 1 and \u0019 = 3. The single particle spectra (right axis, red bars) is smooth, while oscillations appear in co-incidence measurements. See the inset of Figure 3b for a spectrum of the 2nd electron, co-incident with a 12-quanta-loss state of the first electron. In such an electron- pair experiment, energy-gains are unique to the second electron, and hence could be used to record co- incident spectra without separating the paths of the two electrons. Figure 3 – Electron-electron interaction for two distant electrons in a beam, mediated by long-lived photons. The colormap, \u0011\u0012\u0013,\u0015 j+j\u0011 \u0016, is the co-incident probability for \u000fℏ\u0002\u0003 energy-loss of the first electron and \u0017ℏ\u0002\u0003 gain of the second. (a) Strong coupling, \u0019 = 1, allows for mutually exclusive states (dash circle) of the electron pair, where if the first electron loses one quantum, the second cannot be loss-less. (b) Stronger interactions induces rich entanglement features (see inset for lineout of the second electron spectra, co-incident with \u000f = 12). The last part this letter utilizes the above derivations to quantitatively evaluate the coupling constant via PINEM. Specifically, the term ! = \u0019| | in eq. (6) links the coupling constant to the classical acceleration of an electron by the mode’s field. The field is represented as a coherent state | ⟩, and the acceleration or deceleration is represented by !. For an electron traveling the path 0 < l < m near a straight arm of the cavity, the parallel acceleration is given by the light-field component, #n (o)5l, p(l)8, evaluated for time p(l). The superscript \u0002 indicates a frequency dependence. The classically calculated electron-energy gain within the interaction region is q ∫#n (o)5l, p(l)8sl t \u0003 , where q is the electron charge (transverse recoil is negligible, see supplementary section S.1.2.2). ! is then the unit-less ratio between the electron-energy gain and the photon energy (eq. (3) in ref. [10]), which with eq. (6) determines the coupling constant as \u0019 = ! | | = 1 | | q 2ℏ\u0002 u #n (o)5l, p(l)8sl t \u0003 . (9) Although \u0019 is calculated from classical fields, it is a geometrical property of the apparatus for a given electron zero-loss energy, since | | = _⟨\u000f⟩∝#n. Intuitively, one can interpret \u0019 as the strength of a PINEM effect for one cavity photon. Eq. (9) has few important aspects, especially when implemented to a long interaction length, e.g. many µm: (i) The optimal coupling occurs when #n (o)5l, p(l)8 is constant along the electron trajectory, which occurs at perfect phase-matching.(ii) Conceptually, the ultimate coupling would be for a straight fiber with periodic boundary conditions and length m. One can realistically reach 1/√2 of that, when accounting for the backward propagating mode in a realistic cavity. (iii) The coupling bandwidth is limited by dispersion (see Figure 1b). A cavity design for a strong coupling requires a small optical-mode volume, matching the velocity of the electron with that of the mode, and", + "reach 1/√2 of that, when accounting for the backward propagating mode in a realistic cavity. (iii) The coupling bandwidth is limited by dispersion (see Figure 1b). A cavity design for a strong coupling requires a small optical-mode volume, matching the velocity of the electron with that of the mode, and a meaningful field component #n in vacuum. Those can be achieved in WGM cavity, based on a straight single-mode fiber (see Figure 1a). The small fiber diameter plays multiple roles: it minimizes the mode volume, increases the evanescent tails in vacuum that interact with the electron, increase the field component, #n, and pushes the modal phase velocity towards the speed of the relativistic electrons. The stadium-geometry of the proposed resonator allows for a long straight section, while minimizing additional mode volume and losses. Figure 4 shows the coupling of electron beams to a mode with a photon energy of ℏ\u0002\u0003 = 1.1654 (vacuum-wavelength = 1064 \u000f\u0010) in a step index profile [48]. For the selected parameters (fiber diameter of 463 nm, electron zero-loss energy of 200 keV) the coupling reaches \u0019 = 0.76 for 100 µm interaction length. Figure 4b presents the spectral width of the interaction, derived from the coherence length [49]. Comparison to the free spectral range for cavities with total circumference of 2m (dashed line), indicates the number of modes with which the electron may couple. A Si3N4 cavity (orange line) allows for phase-matching with 4 optical modes, while silicon (blue line, for 213 nm diameter), allows for the coupling of a single optical mode, or none. As a concrete example, I consider the stadium-shaped WGM cavity with a 100 µm straight arm and a negligible semicircle circumference, comprising a Si3N4 fiber with a diameter of 463 nm. An electron beam passing close to the cavity wall would have a coupling strength of \u0019 = 0.76/√2 = 0.53, where the √2 accounts for the non-interacting cavity-arm. The field evanescence length 120 nm in vacuum. Thus, an electron beam with a semiconvergence angle of 0.15 mili-radians and a waist of 10 nm would experience a uniform light-field. These are achievable parameters in contemporary electron microscopes. With this design scheme, and possible other modifications, unprecedented strong coupling and long-lived entanglement effects may be reached in the not-distant future. Figure 4 – (a) The coupling constant to a 100-µm-long Si3N4 step-index fiber (left axis, blue) and the phase- matched electron energy (right axis, orange) for 1064 nm photons, as a function of the fiber diameter. This calculation assumes periodic boundary conditions, thus, a realistic coupling would be smaller by √2. (b) The coupling bandwidth (orange) narrows for longer interactions. Increased dispersion (e.g. Si, blue line) can result in a narrower bandwidth than the free spectral range (dash diagonal), which limits the coupling to a single optical mode at most. To conclude, this work proposes a path towards a strong coupling regime between electrons and cavity- photons based on narrow-band phase-matching, and investigates phenomena that this regime may enable. The analytical model addresses EELS and PINEM on an equal footing alongside strong coupling", + "to a single optical mode at most. To conclude, this work proposes a path towards a strong coupling regime between electrons and cavity- photons based on narrow-band phase-matching, and investigates phenomena that this regime may enable. The analytical model addresses EELS and PINEM on an equal footing alongside strong coupling phenomena. The coupling, \u0019, may be retrieved experimentally using PINEM, via eq. (9), or using EELS, via the |\u0019|\u0016 loss probability, per optical mode. Additionally, strong coupling to a cavity can entwine the quantum state of two consecutive electrons, entangling their final energies. 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Yalunin, Hugo Lourenço-Martins, Armin Feist and Claus Ropers for illuminating discussions and support. Supplementary Material: Entanglements of electrons and cavity-photons in the strong coupling regime Ofer Kfir1 1University of Gttingen, IV. Physical Institute, Gttingen, Germany Contents S.1Basics of the coherent interaction between electronic and photonic states 14 S.1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 S.1.2 Electron-dispersion effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 S.1.2.1 Relevant distance for electron-energy dispersion effects . . . . . . . . . . . . . 15 S.1.2.2 Estimation of the electron recoil (transverse deflection) . . . . . . . . . . . . 15 S.1.3 The displacement operator - S-matrix approach . . . . . . . . . . . . . . . . . . . . . 16 S.1.3.1 Derivation of the S-matrix as a displacement operator . . . . . . . . . . . . . 16 S.2Effects for photons interacting with an electron-beam 16 S.2.1 EELS as field-less PINEM - strong interaction without a driving field . . . . . . . . . 16 S.2.2 PINEM - electron interaction with a strong laser-field . . . . . . . . . . . . . . . . . 17 S.2.2.1 Retrieving the experimental PINEM spectrum for weak interactions with strong fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 S.2.2.2 Comments on PINEM with nearly classical fields . . . . . . . . . . . . . . . . 20 S.2.2.3 Separated expressions for gain and for loss channels of electron-photon inter- actions .", + ". . . . . . . . . . 19 S.2.2.2 Comments on PINEM with nearly classical fields . . . . . . . . . . . . . . . . 20 S.2.2.3 Separated expressions for gain and for loss channels of electron-photon inter- actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 S.3Two-electron interaction mediated by cavity photons 22 S.4Ladder operators for the relativistic electron 26 S.5Quantitative evaluation of electron-fiber coupling 27 A Assisting derivations 32 13 S.1 Basics of the coherent interaction between electronic and photonic states S.1.1 Assumptions • The electron state is . |ψ⟩= ∞ X n=0 ∞ X j=−∞ cnj |Ej, n⟩. (1) The electronic part of the state may just be written as |Ej⟩, similar to Feist et al. 2015 [1]. • The ladder operators are ˆb,ˆb†, with the commutation relation h ˆb, ˆb†i = 0. This commutation relation holds throughout relevant the energy spectrum. For fast electrons, \u0010 ˆb \u0011† = ˆb†. For very slow electrons, the lowering and lifting operators are not the hermitian conjugate of each other. See details in section S.4. • The electrons interact with a harmonic system, with energy spacing ω0, such that Ej+k = Ej + kℏω0. • For the photons the ladder operators are the standard ˆa and ˆa†, with \u0002 ˆa, ˆa†\u0003 = 1.(See for example quantization in Scully and Zubairy [2] or Mandel and Wolf [3]) • The operator of the electric field for free space, is ⃗ˆE (r, t) = 1 L3/2 X k X s r ℏω 2ε0 h iˆak,s (0) εk,sei(k·r−ωt) + h.c. i . That can be written for simplicity as ⃗ˆE (r, t) = ⃗ˆE(+) (r, t) + ⃗ˆE(−) (r, t) Here I just abbreviate ⃗ˆE(+) = ˆa, (2) where the time and space dependency are understood. The quantization of fiber modes has a similar form, see section S.5. • The interaction Hamiltonian is He + Hnp + HI. The Hamiltonian for a local interaction is H = r E2 rest + \u0010 ˆPc \u00112 + ℏω0ˆa†ˆa + ξ \u0010 ˆbˆa† + ˆb†ˆa \u0011 . (3) The exact form of the electron Hamiltonian, He, and the operators b, b† is described in section S.4. ξ is the local coupling strength, with units of energy. It is meaningful only in the context of the total interaction strength, as e.g. in eq. (12). 14 S.1.2 Electron-dispersion effects S.1.2.1 Relevant distance for electron-energy dispersion effects The distances that are relevant for dispersion effects can be evaluated from a Taylor expantion of the phase for an electron plane-wave φ(E,z) = φ(E,0) + P(E) z ℏ (4) = φ(E,0) + z ℏc q E2 −E2 rest (5) (E=E0+∆E) ≈ φ(E,0) + z ℏc q E2 0 −E2 rest + E0 (∆E) p E2 0 −E2 rest − E2 rest (∆E)2 2 (E2 0 −E2 rest)3/2 ! . (6) The", + "φ(E,z) = φ(E,0) + P(E) z ℏ (4) = φ(E,0) + z ℏc q E2 −E2 rest (5) (E=E0+∆E) ≈ φ(E,0) + z ℏc q E2 0 −E2 rest + E0 (∆E) p E2 0 −E2 rest − E2 rest (∆E)2 2 (E2 0 −E2 rest)3/2 ! . (6) The dispersion can be neglected for short propagation distances, z ≪zdispersion, where zdispersion = ℏc · E2 rest (∆E)2 2 (E2 0 −E2 rest)3/2 −1 . (7) For example, electrons at 200 keV, and 11.65 eV bandwidth (10 orders of photons with 1064 nm vacuum-wavelength), zdispersion ≈5.3 mm. S.1.2.2 Estimation of the electron recoil (transverse deflection) Since the optical mode has both parallel field (Ez )and transverse field (Ex, Ey) components, it can, in principle, deflect the electron and compromise the validity one-dimentional description, as in this work. For the optical-mode parameters described in this work, say in Figure 4 of the manuscript, this deflection is small. Given a similar field components, as in our case, one can assume for simplicity |Ex| ≈|Ez|. During the interaction time L/v0, the transverse deflection can be estimated by the classical impulse is the added transverse momentum ∆Px = q|Ex|(L/v0). v0 is the group velocity of the relativistic electron. Thus, the final deflection θf from the induced transverse momentum is θf = ∆Px P0 = q|Ex|L v0P0 (8) ≈q|Ez|L v0P0 . (9) using the phase-matched coupling α = qEzL 2ℏω0 , ℏω0 = 1.1eV , P0 = γm0v0, and the parameters for 200 keV electrons, γ = 1.39, and v0 = 0.7c, one can write θf ≈q|Ez|L v0P0 ≈α 2.2eV z }| { 2ℏω0 v0P0 |{z} γm0(0.7c)2 ≈α 2.2eV 338keV ≈α · 6.5e −6. (10) The deflection roughly scales with the coupling, and is only few micro-radians. Assuming the acceleration is constant and the deflection trajectory is parabolic, this FINAL deflection correspond to a trajectory x(z) = θf L z2 2 , which result in a final deflection x(z = L) ≈0.2 nm for α = 1 and L = 100 µm. 15 S.1.3 The displacement operator - S-matrix approach The scattering operator (in the interaction picture), ˆS need to be accounted for in full, to be valid for strong couplings, ˆS =T exp \u0014 −i ℏ Z ∞ −∞ ξ \u0010 ˆbˆa† + ˆb†ˆa \u0011\u0015 (11) = exp \u0014 −iξτ ℏ \u0010 ˆbˆa† + ˆb†ˆa \u0011\u0015 . (12) Here, I removed the time-ordering operator, T , since the there is no time dependence for the operator product ˆbˆa† for interaction lengths short enough to suppress dispersion (see eq. (7)) ˆb(t)ˆa†(t) = ˆbeiωtˆa†e−iωt = ˆbˆa†. τ is an effective interaction duration. Additional phases, such as temporal-delays accumulated by the electron energy-states, can be neglected below the characteristic dispersion distance, as in eq. (7). S.1.3.1 Derivation of the S-matrix as a displacement operator Eq. (12) has the form of the displacement operator, D \u0010 ˆbα \u0011 = exp \u0010 αˆbˆa† −α∗ˆb†ˆa \u0011 , with a substitution −iξτ ℏ= α (13) −iξτ ℏ= −α∗. (14) For this to be correct, ξτ ℏ∈ℜeal. Aditional comments: • The only difference of", + "S-matrix as a displacement operator Eq. (12) has the form of the displacement operator, D \u0010 ˆbα \u0011 = exp \u0010 αˆbˆa† −α∗ˆb†ˆa \u0011 , with a substitution −iξτ ℏ= α (13) −iξτ ℏ= −α∗. (14) For this to be correct, ξτ ℏ∈ℜeal. Aditional comments: • The only difference of D \u0010 ˆbα \u0011 from D (α) is that ˆb is an operator. It implies conservation of energy, where every Fock-state is entangled with the corresponding state of an electron energy loss |Ej−n, n⟩. The splitting of the energy between the electron channel and photon channel resembles a beam-splitter operator, but with ˆbˆa† instead of the ˆa1ˆa† 2 • If ξτ ℏ∈ℜeal, than the translation parameter, α, is purely imaginary. This means that when the electrons interact with radiation they change momentum, without instantaneous shifts. Time (or propagation) translates momentum difference to modifications of the probability distribution. S.2 Effects for photons interacting with an electron-beam S.2.1 EELS as field-less PINEM - strong interaction without a driving field A strong interaction depends on the coupling parameter, α, with the light field acting only as the initial state. Due to conservation of energy, one expects to end with energy-loss, E−k, with k > 0. 16 The probability amplitude to find an electron in energy E−k can be written as ⟨E−k, n| D(ˆbα) |E0, 0⟩ BCH = e |α|2 2 ⟨E−k, n| e−α∗ˆb†ˆaeαˆbˆa† |E0, 0⟩ (15) = e |α|2 2 ⟨E−k, n| ∞ X m,ℓ=0 (−α∗)m \u0010 ˆb†\u0011m ˆam m! αℓˆbℓ\u0000ˆa†\u0001ℓ ℓ! |E0, 0⟩ (16) = e |α|2 2 ∞ X m,ℓ=0 ⟨E−k−m, n + m| p (n + m)! √ n! (−α∗)m m! αℓ ℓ! √ ℓ! |E−ℓ, ℓ⟩. (17) Here, I used the Backer-Campbell-Hausdorffformula (BCH) to express the displacement operator(eq. (128)), and used \u0000a†\u0001m |n⟩= q (n+m)! n! |n + m⟩. Using the orthogonality of the electron-energy states and the photon states, one get ⟨E−k−m, n + m|E−ℓ, ℓ⟩= δ−k−m,−ℓδn+m,ℓ, so n = k (conservation of energy) and ℓ= m + n. So, the above is = e |α|2 2 ∞ X m=0 p (n + m)! √ n! (−α∗)m m! αmαn (n + m)! p (n + m)! (18) = e |α|2 2 αn √ n! ∞ X m=0 \u0010 −|α|2\u0011m m! = e−|α|2 2 αn √ n! e−|α|2 (19) = e−|α|2 2 αn √ n! = e−|α|2 2 αk √ k! . (20) This is the expected Poisson distribution (see ref. [4]) Pk = ⟨E−k, k| D(ˆbα) |E0, 0⟩ 2 = e−|α|2 α2k k! (21) Typical EELS is retrieved for |α|2 ≪1, RightarrowP0 ≈ \u00001 −|α|2\u0001 , P1 = |α|2. S.2.2 PINEM - electron interaction with a strong laser-field For PINEM, the initial state |ψi⟩, before the electron interacts with light is an uncorrelated state, |ψi⟩= |E0⟩⊗|β⟩= |E0, β⟩. For large β the optical coherent state is a good approximation for classical fields. The important quantum numbers for the final state are the final quanta of electron-energy gain, k, and the remaining number of photons n, ψPINEM f = ∞ X n=0 ∞ X k=−∞ cn,k |Ek, n⟩. The PINEM interaction", + "β the optical coherent state is a good approximation for classical fields. The important quantum numbers for the final state are the final quanta of electron-energy gain, k, and the remaining number of photons n, ψPINEM f = ∞ X n=0 ∞ X k=−∞ cn,k |Ek, n⟩. The PINEM interaction can be written by the displacement operator. cn,k = ⟨Ek, n| D(ˆbα) |E0, β⟩ BCH = (22) = e |α|2 2 ⟨Ek, n| e−α∗ˆb†ˆaeαˆbˆa† |E0, β⟩ (23) = e |α|2 2 ⟨Ek, n| ∞ X m,ℓ,j=0 (−α∗)m \u0010 ˆb†\u0011m ˆam m! | {z } e−α∗ˆb†ˆa αℓˆbℓ\u0000ˆa†\u0001ℓ ℓ! | {z } eαˆbˆa† e−|β|2 2 βj √j! | {z } |β⟩ |E0, j⟩ (24) = e |α|2−|β|2 2 ∞ X m,ℓ,j=0 ⟨Ek−m, n + m| r (n + m)! n! (−α∗)m m! αℓ ℓ! βj √j! s (j + ℓ)! j! |E−ℓ, j + ℓ⟩. (25) 17 Orthogonality of the states imposes ⟨Ek−m, n + m|E−ℓ, j + ℓ⟩= δk−m,−ℓδn+m,j+ℓ (26) so, m = k + ℓ, n + m = n + k + ℓ= j + ℓ. (27) Thus, one remain with a summation over ℓ, cn,k = e |α|2−|β|2 2 ∞ X ℓ=0 r (n + k + ℓ)! n! (−α∗)k+ℓ (k + ℓ)! αℓ ℓ! βn+k (n + k)! p (n + k + ℓ)!. (28) After some rearrangements, the final expression for the final-state amplitudes is cn,k = e |α|2−|β|2 2 (−α∗)k βn+k √ n! ∞ X ℓ=0 (n + k + ℓ)! (n + k)! | {z } (∗) \u0010 −|α|2\u0011ℓ (k + ℓ)!ℓ! . (29) This is an exact expression for the quantum state following PINEM, at any coupling strength. To extend this expression for gain (k > 0) and for loss (k < 0) one can replace the factorial operations by Riemann’s gamma function x! →Γ (x + 1), which diverges for negative integers. Since possible negative factorials terms of (k + ℓ)!ℓ! diverge in the denominator, their corresponding arguments can be ignored. For (n + k) < 0, the term marked (∗), is either (∗) = 1 for ℓ= 0, or (∗) = 0 for ℓ> 0, and is thus regularized. Section S.2.2.3 retrieves the explicit PINEM coefficients for gain and loss, using the factorials of explicitly positive integers, rather than Riemann’s Gamma-function. One example of PINEM-like spectrum is in the main text (figure 2c), and here, figureS.1 presents similar spectrograms, for various coupling constants. 18 Figure S.1: The electron-photon spectrogram for various coupling constants, assuming an initial cavity population of 100 photons in the form of a coherent state |β = 10⟩. The colormap is the co-incidence of a particular energy combination |cn,k|2 of the photon and electron. The bottom axis is the initial (dashed line) and final (blue bars) distribution of the Fock-states for photons in the cavity. The right axis (red bars) is the electron spectrum S.2.2.1 Retrieving the experimental PINEM spectrum for weak interactions with strong fields Here, I show how the derivation above retrieves the known PINEM spectrum for the electron, and in what conditions the field is decoupled from", + "for photons in the cavity. The right axis (red bars) is the electron spectrum S.2.2.1 Retrieving the experimental PINEM spectrum for weak interactions with strong fields Here, I show how the derivation above retrieves the known PINEM spectrum for the electron, and in what conditions the field is decoupled from the electron state modification. Using eq. (29), in the parameter regime accessible to experiments to date, this derivation should yield the a Bessel-function spectrum - ∝|Jk (2 |g|)|2, with a possible additional phase. In the experiments the coupling is weak, the field is strong, and there are only few quanta of energy exchanged between the photons and the electrons |α|2 ≪1 (30) ⟨n + k⟩= ⟨j⟩= |β|2 (31) ℓ⪅k ≪n. (32) Although the summation is up to ℓ→∞, the argument in the sum decays rapidly for ℓ> |αβ|2, so one can compare ℓwith other parameters of this system. The comparison to |βα| becomes clearer from eq. (34) and the definition of the Bessel-function of the first kind. By employing eq. (134), and the ratio ℓ/ (n + k) ≪1, one can write (∗) ≈(n + k)ℓ. In that case, the summation arguments 19 acquire the following form (n + k + ℓ)! (n + k)! \u0010 −|α|2\u0011ℓ (k + ℓ)!ℓ! ≈(n + k)ℓ \u0010 −|α|2\u0011ℓ (k + ℓ)!ℓ! (33) = \u0010 − α √ n + k 2\u0011ℓ (k + ℓ)!ℓ! (34) = \u0010 −|g|2\u0011ℓ (k + ℓ)!ℓ! , (35) where g is g = α √ n + k ≈α |β| . (36) This brings the form of the Bessel-function amplitudes to the energy spectrum, cn,k = e ≪1 z}|{ |α|2 −|β|2 2 βn+k √ n! eik arg(−α∗) \u0012 2 |g| 2 √ n + k \u0013k ∞ X ℓ=0 \u0010 −(2|g|)2 4 \u0011ℓ (k + ℓ)!ℓ! (37) = e−|β|2 2 βn+k √ n! p (n + k) −keik arg(−g∗)Jk (2 |g|) . (38) The above approximation almost reproduces the Bessel-like amplitudes of PINEM, but it leaves some correlations between n and k. To remove these correlations and retrieve classical-field effects, one has to neglect correlations in the coherent state. Specifically when assuming < j >=< n+k >≈|β|2 and √ n + k ≈ p ⟨n + k⟩the following is simplified \u0010√ n + k \u0011−k ≈ \u0010 βe−i arg(β)\u0011−k = β−keik arg(β) (39) βn+k \u0010√ n + k \u0011−k | {z } |β|−k ≈βn · eik arg(β). (40) The photon states and electron states are now separable. |ψf⟩= X n,k cn,k |Ek, n⟩ (41) ≈ \"X n e −|β|2 2 βn √ n! #   X k eik =arg(βg) z }| { (arg (β) + arg (−g∗))Jk (2 |g|) |Ek, n⟩   (42) = |β⟩⊗ X k h eik(arg(βg))Jk (2 |g|) i |Ek⟩ (43) = |β⟩⊗ X k ck |Ek⟩. (44) I used here the relation g = −g∗, or arg (g) = arg (−g∗) since α = −α∗. S.2.2.2 Comments on PINEM with nearly classical fields Some points from the above derivation of final state for strong fields interacting weakly with electrons are worth stressing: 20 •", + "ck |Ek⟩. (44) I used here the relation g = −g∗, or arg (g) = arg (−g∗) since α = −α∗. S.2.2.2 Comments on PINEM with nearly classical fields Some points from the above derivation of final state for strong fields interacting weakly with electrons are worth stressing: 20 • g has the same meaning as for classical fields, as in Refs. [1, 5]. • The electronic states have the amplitude as in the experiments, ck = eik(arg(βg))Jk (2 |g|) , (45) not only the probabilities. • g is proportional to the electric field and the coupling constant, g ∝α |E|, since |E| ∝ p ⟨n + k⟩= |β|. This is in agreement with the it’s classical definition. • There is a phase locking between the initial coherent state and the electron state. It appears in the argument eik(arg(βg)). • The locking phase just contributes an linear phase with the energy, that is, it provides for the definition of time-zero. • |g| = |αβ|, which means that one can increase the width of the electron spectrum (have many PINEM orders, ∆E ∝2 |g|). The scaling, for a given cavity, will be linear with the interaction length (via α), and linear with the PINEM-driving electric field (via β). • Since for a coherent state |β⟩, the coupling and g are related by g = α|β|, it means that the coupling, α can be retrieved from classical calculations of PINEM by α = g |β|. (46) • As mentioned in the main text, the equivalence of the gain and loss channel originates from a small coupling, |α| ≪1. The mean energy loss is |α|2 in any experimental configuration, EELS, PINEM, weak- or strong-coupling. • In practice, the correlations between the photon states n and the electron energy indices k is negligible for a high-β coherent state and weak coupling. This is visualy clear from the calculation in the main text. S.2.2.3 Separated expressions for gain and for loss channels of electron-photon inter- actions One reason to keep the factorials in eq.(29) is the numerical errors induced when evaluating factorials x! through the Riemann Gamma function Γ(x+1). Thus, for the calculation presented in the figures of the main text, I explicitly separated the expression to the cases of k ≥0 and k < 0. This is done by choosing the summation index that spans 0 →∞. That is, ℓfor k ≥0 and m for k < 0. The other index is eliminated by the substitution for k ≥0 ⇒m = k + ℓ (47) for k < 0 ⇒ℓ= m −k. (48) Such a separation would explicitly assure that the physical constraint are met m, ℓ, j, n ≥0. First, for k ≥0 we have eq. (29), with ℓas the summation index. For k < 0, the index selection in in eq. (48) is ℓ= m −k and n = j −k, cn,k (k<0,n≥0) = e |α|2−|β|2 2 (−α∗)k βn+k √ n! ∞ X m=0 (n + m)! (n + k)! \u0010 −|α|2\u0011m−k m! (m −k)! (49) = e |α|2−|β|2 2 α−kβn+k (n + k)!", + "0, the index selection in in eq. (48) is ℓ= m −k and n = j −k, cn,k (k<0,n≥0) = e |α|2−|β|2 2 (−α∗)k βn+k √ n! ∞ X m=0 (n + m)! (n + k)! \u0010 −|α|2\u0011m−k m! (m −k)! (49) = e |α|2−|β|2 2 α−kβn+k (n + k)! √ n! ∞ X m=0 (n + m)! \u0010 −|α|2\u0011m m! (m −k)! (50) = e |α|2−|β|2 2 α|k|βn+k (n + k)! √ n! ∞ X m=0 (n + m)! \u0010 −|α|2\u0011m m! (m + |k|)! (51) 21 The last expressions are retrieved by changing t −k →|k|, and using α = −α∗. Thus, the two expressions for gain and loss, eqs. (29) and (51), respectively, can be combined ψP INEM f = ∞ X n=0 ∞ X k=−∞ cn,k |Ek, n⟩ (52) cn,k = e |α|2−|β|2 2 α|k|βn+k (n + k)! √ n! ∞ X ℓ=0 \u0010 −|α|2\u0011ℓ (ℓ+ |k|)!ℓ! · \u001a (n + k + ℓ)! for k ≥0 (n + ℓ)! for k < 0 . (53) Note that for the k < 0 part, I just wrote the arbitrary summation index as ℓinstead of m, and used −k = |k|, while for the k > 0 part, I substituted k = |k|. S.3 Two-electron interaction mediated by cavity photons After an interaction of an electron with a cavity, eq. (20) calculates the final state amplitudes, |ψ⟩= X s e−|α1|2 2 αs 1 √ s! |E−s, s⟩. (54) It is a coherent photonic state with parameter α1, the strength of the first interaction, and an electronic part that conserves a net energy E0. I derive the interaction strength as α1 and α2 for the first and second electron, respectively, to be able to separate their contributions. Typically, one can take equal interaction strengths, α1 = α2, as done in the main text. The loss index is changed here to (s) to differ the loss of the first electron from the gain index of the second electron, k and the photons’ index n. In the manuscript the index n instead of s for brevity. I now consider a second electron with energy E0 = E0. The different symbol just marks a difference between the first and the second electrons. Thus, the initial electron-electron-photon state, before the second electron interacts with stored photons is |ψi⟩= X s e−|α1|2 2 αs 1 √ s! |E−s, E0, s⟩. (55) The final state can be characterized by the individual electron-electron-photon states, ψe−e f E = X s≥0, k cs,k |E−s, Ek, n⟩. The coefficients are given by the projection cs,k = ⟨E−s, Ek, n| D(ˆbα2) |ψi⟩ (56) = ⟨E−s, Ek, n| X s,j e−|α1|2 2 αs 1 √ s! D(ˆbα2) |E−s, E0, s⟩ (57) = ⟨E−s, Ek, n| X s,j e−|α1|2 2 αs 1 √ s! e |α2|2 2 X m,ℓ (−α∗ 2)m \u0010 ˆb†\u0011m ˆam m! | {z } e−α∗ 2ˆb†ˆa αℓ 2ˆbℓ\u0000ˆa†\u0001ℓ ℓ! | {z } eα2ˆbˆa† |E−s, E0, s⟩ (58) = X m,ℓ ⟨E−s, Ek−m, n + m| X s,j e−|α1|2 2 αs 1 √ s! e |α2|2 2 r (n", + "√ s! e |α2|2 2 X m,ℓ (−α∗ 2)m \u0010 ˆb†\u0011m ˆam m! | {z } e−α∗ 2ˆb†ˆa αℓ 2ˆbℓ\u0000ˆa†\u0001ℓ ℓ! | {z } eα2ˆbˆa† |E−s, E0, s⟩ (58) = X m,ℓ ⟨E−s, Ek−m, n + m| X s,j e−|α1|2 2 αs 1 √ s! e |α2|2 2 r (n + m)! n! (−α∗ 2)m m! · (59) αℓ 2 ℓ! r (s + ℓ)! s! |E−s, E−ℓ, s + ℓ⟩. (60) 22 Note that here the operator ˆb is acting on the second electron, leaving the first unchanged. The states’ orthogonality imposes ⟨E−s, Ek−m, n + m|E−s, E−ℓ, s + ℓ⟩= δn+m,s+ℓδm−k,ℓ (61) ⇒m = ℓ+ k (62) s = n + k. (63) Inserting the indices selection, and using the states’ orthogonality, eq. (59) is = ∞ X ℓ=0 e−|α1|2 2 αs 1 √ s! e |α2|2 2 r (n + ℓ+ k)! n! (−α∗ 2)ℓ(−α∗ 2)k (ℓ+ k)! αℓ 2 ℓ! s (n + ℓ+ k)! (n + k)! (64) = e−|α1|2 2 αs 1 √ s! e |α2|2 2 (−α∗ 2)k p n! (n + k)! ∞ X ℓ=0 (n + ℓ+ k)! \u0010 −|α2|2\u0011ℓ (ℓ+ k)!ℓ! (65) . (66) The relation n + k = s allows the last equation to be written with as a function of the electron energies only, k, s. So, one can write the coefficients cs,k cs,k>0 = e−|α1|2 2 αs 1e |α2|2 2 (−α∗ 2)k p (s −k)! ∞ X ℓ=0 (ℓ+ s)! s! \u0010 −|α2|2\u0011ℓ (ℓ+ k)!ℓ! . (67) Factorials of negative numbers diverge according to the Riemann’s Gamma function. The term p (s −k)! ≥0 diverges for s < k. In other words, the highest k is the full conversion energy taken from the 1st electron to the 2nd. Thus, this term nullifies the probability for a non-physical energy- gain of the second electron. Note that in the main text I chose to simplify the system by choosing α1 = α2 = α, which is a realistic realization of eq. (67), when the two electrons share the same beam path and interact with the same cavity. Similar to the discussion in section S.2.2.3, the indices selection m, ℓdiffers for the gain- and loss- channels, as used in practice to calculate the 2-particle amplitudes numerically, with the substitutions m = ℓ+ k for k ≥0 (68) ℓ= m −k for k < 0. (69) First, for the case of energy gain by the second electron, k > 0, the two-electron probabilty amplitudes are in eq. (67). For the case of energy loss by the second electron, k < 0, the proper index to keep is m, with the relation ℓ= m −k = m + |k| from eq. (69). = ∞ X m=0 e−|α1|2 2 αs 1 √ s! e |α2|2 2 r (n + m)! n! (−α∗ 2)m m! αm 2 α|k| 2 (m + |k|)! s (n + m)! (s)! (70) = e−|α1|2 2 αs 1e |α2|2 2 α|k| 2 p (s −k)! ∞ X m=0 (m + (s + |k|))! s! \u0010 −|α2|2\u0011m (m + |k|)!m!. (71)", + "|α2|2 2 r (n + m)! n! (−α∗ 2)m m! αm 2 α|k| 2 (m + |k|)! s (n + m)! (s)! (70) = e−|α1|2 2 αs 1e |α2|2 2 α|k| 2 p (s −k)! ∞ X m=0 (m + (s + |k|))! s! \u0010 −|α2|2\u0011m (m + |k|)!m!. (71) Finally, one can combine the expressions for the energy-gain and energy-loss for the second electron, ψe−e f E = X s X k≤s cs,k |E−s, Ek, s −k⟩ (72) cs,k = e−|α1|2 2 e |α2|2 2 αs 1α|k| 2 p (s −k)!    P∞ ℓ=0 (ℓ+s)! s! (−|α2|2) ℓ (ℓ+k)!ℓ! for s ≥k ≥0 P∞ m=0 (m+(s+|k|))! s! (−|α2|2) m (m+|k|)!m! for k < 0 (73) 23 Using the equality −α∗ 2 = α2, setting |k| appropriately, and using just either ℓas a summation index, a more compact equation can be written cs,k = e−|α1|2 2 αs 1 s! e |α2|2 2 α|k| 2 p (s −k)! ∞ X ℓ=0 \u0010 −|α2|2\u0011ℓ (ℓ+ |k|)!ℓ! · \u001a (ℓ+ s)! for s ≥k ≥0 (ℓ+ s + |k|)! for k < 0 . (74) Except for the spectrograms in the main text, for the coefficients cs,k I added some here, for different coupling strength, α = α1 = α2 Figure S.2: The two-electron spectrogram for various values of equal coupling constants. The colormap is the co-incidence of a particular energy comibination |cs,k|2. The bottom axis (blue bars) is the spectrum of the first electron (including only loss channels) and the right axis (red bars) is the spectrum of the second electorn. Note that in the text, the loss index of the first electron is n for brevity, while here it is s. There are several interesting examples to consider 1. Checking the limit of no initial gain – if the interaction strength of the first electron is nullified, |α1| = 0, the interaction of the second electron should result in the spectrum of a single interaction. Only the coefficients with s = 0 survive due to the factor αs 1, and the √ s −k term suggests that k ≤0. The second electron therefore, populates no gain states. According to eq. 24 (74), the coefficients cs,k will be c0,k<0 = e |α2|2 2 1 p (−k)! α−k 2 ∞ X m=0 (m + (−k))! (m −k)! \u0010 −|α2|2\u0011m m! (75) = e −|α2|2 2 α|k| 2 p |k|! , (76) which is a Poissonian probability distribution, just as in eq. (20) . 2. The coefficients retrieved from the two-electron interaction (eq. (74)), and for the PINEM interaction (eq. (53)) are equivalent, by just selecting indices. Since the first electron in the two-electron case induces a coherent state, the following stage, which is the interaction of a coherent state with an approaching electron is identical to strong-coupling PINEM. However, the important difference is the quantum numbers. For PINEM, the quantum numbers (in which one may search for entanglements) are n, k, while for two electrons, their energy states, s, k are important. Thus, for physically relevant purposes, the two cases are sheared n", + "is identical to strong-coupling PINEM. However, the important difference is the quantum numbers. For PINEM, the quantum numbers (in which one may search for entanglements) are n, k, while for two electrons, their energy states, s, k are important. Thus, for physically relevant purposes, the two cases are sheared n = s−k. Such a shear can be identified by comparing figure 2 and figure 3 in the main text. One can also consider the 1st electron spectrum as corresponding to the initial optical state, which differs from the final, non-coherent-state-like photon distribution. 3. Approaching strong-field PINEM for the 2nd electron. Similar to section S.2.2.2, cs,k can resemble the experimentally measured PINEM for weak coupling limit. For the two-electron case, that requires verri different coupling strengths, |α1 ≫1 ≫|α2||.For a large energy deposition in the cavity by the first electron, one can assume (ℓ+ |k|) , ℓ≪s, and approximate (ℓ+ s + |k|)! ≈s!sℓs|k| (77) ⇒(ℓ+ s + |k|)! \u0010 −|α2|2\u0011ℓ = s!s|k| \u0010 − α2 √s 2\u0011ℓ . (78) The factorial approximation is justified in eq. (134). Considering the gain/loss dependent part of (74), including the term p (s −k)!, and changing k to |k| in a consistent manner for k < 0 and k ≥0, one can write the following equalities to combine the gain and loss parts ( (ℓ+ s)! ((s −|k|)!)−1 2 (ℓ+ s + |k|)! ((s + |k|)!)−1 2 = ( s!sℓ(s!)−1 2 s |k| 2 for s ≥k ≥0 s!sℓs|k| (s!)−1 2 s−|k| 2 for k < 0 (79) = √ s! \u0000√s \u0001|k| sℓ, (80) Incorporating that into eq. (74) gives a separable equation. Same logic here applied as cs,k = \u0014 e−|α1|2 2 αs 1 √ s! \u0015   e |α2|2 2 \u0000α2 √s \u0001|k| ∞ X ℓ=0 \u0010 −|α2 √s|2\u0011ℓ (ℓ+ |k|)!ℓ! | {z } Jk(2|g|)   , (81) where, again, g = α2 √s ≈α2 |α1|, and this retrieves the experimental Bessel-amplitudes of PINEM. One should note that the above separability is naturally occurring for the quantum numbers s, k since the first electron is unaffected by any detail of the intaraction with a second elctron. It already left the interaction region. In the PINEM case, where the final state is expressed with n, k and s = n + k, the two states cannot be separated. A similar treatment leading to eq. (42) would result in cn,k = h e−|β|2 2 βn+k √n+k i [· · · ], which is clearly not separable. For this reason, the quantum fluctuations remain here, while they have to be mitigated in eq. (42). 25 S.4 Ladder operators for the relativistic electron • Assuming the energy is allowed in levels |n⟩, the Hamiltonian comply with ˆH |n⟩= En |n⟩. • In this section, n is the energy level-index of the electron, with respect to the zero-loss energy En=0 = E0. The number operator is ˆn = ˆH −E0 ℏω0 . For the zero-loss energy, ˆH |0⟩= E0 |0⟩. E0 relates to the electron rest energy Erest and the zero-loss momentum P0 by En=0 =", + "is the energy level-index of the electron, with respect to the zero-loss energy En=0 = E0. The number operator is ˆn = ˆH −E0 ℏω0 . For the zero-loss energy, ˆH |0⟩= E0 |0⟩. E0 relates to the electron rest energy Erest and the zero-loss momentum P0 by En=0 = q E2 rest + (P0c)2. • For nearly plane-wave electrons, the ladder operators commute [b†, b] = 0. In that case, They cannot construct the Hamiltonian. The Hamiltonian is, by definition, sensitive to the level index, and hence cannot commute with a ladder operator, e.g. ˆH \u0000b† |n⟩ \u0001 = En+1 \u0000b† |n⟩ \u0001 ̸= b† \u0010 ˆH |n⟩ \u0011 = Enb† (|n⟩). • The momentum of state |n⟩is P0 + Pn. Thus, it can be written as |n⟩= exp \u0002 i ℏ(P0 + Pn) \u0003 • The dispersion relation for the electrons around the zero-loss energy is E = q E2 rest + (P0 + Pn)2 c2 (82) = q E2 rest + (P0c)2 s 1 + 2P0Pnc2 + P 2nc2 E2 rest + (P0c)2 (83) ≈ q E2 rest + (P0c)2 1 + 1 2 2P0Pnc2 + P 2 nc2 E2 rest + (P0c)2 ! (84) =const + P0Pnc2 q E2 rest + (P0c)2 + P 2 nc2 2 q E2 rest + (P0c)2 (85) =EZero−loss + vZero−loss · Pn \u0012 1 + 1 2 Pn P0 \u0013 (86) • For relativistic electrons, Pn ≪P0, the Hamiltonian is linear with the momentum, Pn. The ladder operators can be written explicitly as ˆb† = ei∆kx and ˆb = e−i∆kx, with ∆k = (Pn+1 −Pn) /ℏ= ω0/vZero−loss. • To show the ladder operators are correct, one needs to show is that ˆH \u0000b† |n⟩ \u0001 = En+1 \u0000b† |n⟩ \u0001 ˆH \u0000b† |n⟩ \u0001 = ˆH \u0010 ei ω0 c x · ei(P0+Pn)x\u0011 (87) = ˆH · ei(P0+Pn+ ω0 c )x (88) = ˆH · ei(P0+Pn+1)x (89) =En+1ei(P0+Pn+1)x (90) =En+1 \u0010 ei ω0 c x · ei(P0+Pn)x\u0011 = En+1 \u0000b† |n⟩ \u0001 . (91) Similarly, ˆH (b |n⟩) = En−1 (b |n⟩). 26 • Since ˆb,ˆb† are pure phasors, they reconstruct the relations in Ref. [1], b† |n⟩= |n + 1⟩ (92) b |n⟩= |n −1⟩. (93) This relation applies also for non-relativistic electrons if the underlying assumptions hold. • Alternatively, the ladder operators can be constructed in a diagonal form ˆb = X n ei(kn−1−kn)x |n⟩⟨n| (94) ˆb† = X n ei(kn+1−kn)x |n⟩⟨n| ., (95) based on the known values of the wave-vectors kn. One can see that, especially for slow electrons ˆb and ˆb† are not exactly hermitian conjugates of each other. Hence, they may not be convenient for the representation of observables quantities. However, even at acceleration voltages of few keV, the dispersion becomes linear enough to allow the assumption that ˆb and ˆb† are complex conjugates. S.5 Quantitative evaluation of electron-fiber coupling This section is mostly technical, in the form of bullet-points that allows, with the sources to follow the quantitative estimation of the coupling constant (say, per 1µm). It is based on Refs. [6, 7] and assisted", + "assumption that ˆb and ˆb† are complex conjugates. S.5 Quantitative evaluation of electron-fiber coupling This section is mostly technical, in the form of bullet-points that allows, with the sources to follow the quantitative estimation of the coupling constant (say, per 1µm). It is based on Refs. [6, 7] and assisted by Prof. Elias N. Glytsis notes about Cylindrical Dielectric Waveguides, 2017. Numerical results from finite-elements calculation are in good agreement with the analytic calculation below. (see figure S.4) In short, I calculate the field distribution for an HE11 mode in a round , clad-less, step-index fiber, and estimate the field close to it’s surface, in vacuum. For a given fiber length of 1 µm, I calculate g and employ eq. (46), to calculate g per photon, that is, the coupling constant α. The average number of photons in the classical field of the mode is evaluated as < n >= U/ℏω, where U is the total field’s energy, per µm, and ℏω is the photon energy. I start with the basic form of the mode. An HE11 mode is always guided in a fiber. It is typically given by the the electric and magnetic fields parallel to the fiber , Ez and Hz EHE11 z (r, φ, z, t) = eiωt−iβzsin φ ( A1J1 \u0000u r a \u0001 r ≤a B1K1 \u0000w r a \u0001 r > a (96) HHE11 z (r, φ, z, t) = eiωt−iβzsin φ ( F1J1 \u0000u r a \u0001 r ≤a G1K1 \u0000w r a \u0001 r > a (97) u = q k2 in −β2 , kin = 2π λ ncore (98) w = q β2 −k2 out , kout = 2π λ nclad=vacuum. (99) Jℓ(x), and Kℓ(x) are the Bessel function of the first kind and the modified Bessel function of the second kind. An electron traversing parallel to the fiber will excite one linearly polarized mode, thus, radial function is sin (φ). The normalization for the radial function is already included in the calculation of A1. First, I note that Ez is the most relevant field component, as it determines g by accelerating or decelerating the co-propagating electrons. For HE11, find the smallest solution of the propatation constant, β, from the equation for ℓ= 1. \u0014 1 u J′ ℓ(u) Jℓ(u) + 1 w K′ ℓ(w) Kℓ(w) \u0015 \"\u0012ncore nclad \u00132 1 u J′ ℓ(u) Jℓ(u) + 1 w K′ ℓ(w) Kℓ(w) # = \u0012 βℓ kout \u00132 \u0014 1 u2 + 1 w2 \u00152 . (100) 27 Here, J′ ℓ(u) = d dxJℓ(x) x=u, and similarly for K′ ℓ(w). From β, one finds u and w. The remaining coefficients, B1, F1, G1 (assuming an arbitrary A1 = 1 for simplicity) are, A1 = 1 (101) B1 = Jℓ(u) Kℓ(w)A1 (102) G1 = Jℓ(u) Kℓ(w)F1 (103) F1 = 1 µ0ω (iβℓ) \u0012 1 u2 + 1 w2 \u0013 \u0014 1 u J′ ℓ(u) Jℓ(u) + 1 w K′ ℓ(w) Kℓ(w) \u0015−1 A1. (104) Now one has a full expression for the mode’s fields. The next step is to find the number of photons ⟨n⟩per", + "Kℓ(w)F1 (103) F1 = 1 µ0ω (iβℓ) \u0012 1 u2 + 1 w2 \u0013 \u0014 1 u J′ ℓ(u) Jℓ(u) + 1 w K′ ℓ(w) Kℓ(w) \u0015−1 A1. (104) Now one has a full expression for the mode’s fields. The next step is to find the number of photons ⟨n⟩per µm. Once one calculates g for classical field, the coupling constant is quantitatively retrieved from the classical-field calculation by |alpha| = |g| |β| = |g| √ ⟨n⟩. In practice, one can choose to normalize A1 per photon so that ⟨n⟩= 1 results (just for the simplicity of the calculation, not for that actual physical case) in the direct form α = g. For a propagating mode, the energy is time-stationary and azimutally uniform, so only the radial distribution requires calculation, at a given time. I choose the time of maximal Ez, along the axis φ = 0. Thus, one can ignore field components that nullify along the axis of φ = 0, or those with a temporal phase shift i, since their quarter-cycle shift nullifies when Ez maximal. The Field components other than Ez are Er = − iβ k2 0n2 −β2 \u0014 ∂rEz + µ0ω βr ∂φHz \u0015 , out-of-phase in time or φ (105) Eφ = − iβ k2 0n2 −β2 \" 1 r ∂φEz −µ0ω β ∂rHz # , i∂rHz is in phase (106) Hr = − iβ k2 0n2 −β2 \u0014 ∂rHz −ε0n2ω βr ∂φEz \u0015 , i∂rHz is in phase (107) Hφ = − iβ k2 0n2 −β2 \u00141 r ∂φHz + ε0n2ω β ∂rEz \u0015 , i∂rHzout-of-phase in time or φ (108) Hz , see eq. (97) Hz is out-of-phase temporally. (109) The in-phase components, spatially and temporally are boxed. The others do not contribute. It is convenient to express the energy in terms of ⃗E and \u0010 µ0ω ⃗H \u0011 , since the factor µ0ω comes either from the above ratios or from eq. (104). The energy is U = 1 2 Z space \u0010 ⃗E · ⃗D + ⃗B · ⃗H \u0011 = 1 2 Z space ε0n2 ⃗E 2 + µ0 \u0012 1 µ0ω \u00132 µ0ω ⃗H 2 ! . (110) Using the relations ε0 = \u0000µ0c2\u0001−1, one can write U = 1 2ε0 Z space \u0012 n2 ⃗E 2 + \u0010 c ω \u00112 µ0ω ⃗H 2\u0013 . (111) To normalize the fiber mode across some volume, we choose a fiber length L [µm], with periodic boundary conditions, to allow for a unidirectional mode. This simplification can be easily taken into account in the cavity design, using the cavity effective length an any particular geometry. For example, the mode effective volume would be larger by a factor of √ 2 for a cavity encapsulated between two mirrors. I assume that the energy distribution is independent of φ. Using these fields, one can require that A1 normalizes the energy to that of one photon, U = A2 1 · L · ε0 Z 2π 0 dθ Z ∞ r=0 \u0012 n2 \u0000E2 z + E2 φ \u0001 + \u0010 c ω \u00112 (µ0ωHr)2", + "distribution is independent of φ. Using these fields, one can require that A1 normalizes the energy to that of one photon, U = A2 1 · L · ε0 Z 2π 0 dθ Z ∞ r=0 \u0012 n2 \u0000E2 z + E2 φ \u0001 + \u0010 c ω \u00112 (µ0ωHr)2 \u0013 rdr != ℏω, (112) 28 where one assumes the fields above were initially scaled according to eq. (101). Thus, A1 is given by A1 = r ℏω 2πε0L \u0014Z ∞ r=0 \u0012 n2 \u0000E2 z + E2 φ \u0001 + \u0010 c ω \u00112 (µ0ωHr)2 \u0013 rdr \u0015−1 2 . (113) This integral is evaluated for the inner and outer segments, R a 0 and R ∞ a using (J1(), ncore) and (K1(), nclad = 1) , respectively. At this point one has the classical field of an HE11 fiber mode with an average energy of one photon. To evaluate the coupling constant, one only needs to calculate g via its definition in refs. [1, 5] g = q 2ℏω Z L 0 Ez (r, φ = 0, z, t(z)), (114) for the electron trajectory (z, t(z)) (see main text). Only Ez(r > a) is relevant to accelerate/decelerate an electron in vacuum. Ez (r > a, φ, z, t) = B1K1 \u0010 w r a \u0011 ei(ωt−βz) (115) B1 = A1 J1(u) K1(w) (116) . (117) The maximal relevant field is, available for electron coupling right at the fiber edge is Ez \u0000a+, 0, 0, 0 \u0001 =B1K1(w) (118) . (119) For a phase-matched interaction the electron experience a time-independent field along its path (z, t(z)), so E (r, φ, z, t(z)) = E (r, φ, 0, 0) . Thus, the maximal PINEM interaction per photon, is on the surface of the fiber, under conditions of full-phase-matching is gmax PINEM per photon = 1 2ℏω Z L 0 Ez \u0000a+, 0, z, t(z) \u0001 dz = qEz (a+, 0, 0, 0) 2ℏω L. (120) This is, quantitatively, the maximum electron-photon coupling αmax = g p ⟨n⟩ = qEz (a+, 0, 0, 0) 2ℏω L . (121) One should note that the maximal coupling scales with the cavity length as αmax ∝ √ L , since the volume normalization included in A1, scales as 1/ √ L. For a step index fiber of Si3N4-core in vacuum having length of 100 µm with periodic boundary conditions, I calculated the coupling properties vs. the fiber diameter (figure S.3). The calculated properties are the optimal coupling, the distance for e−1 decay of the field, the acceleration voltage for phase-matched electrons, and on the right axis, the coherence lengths for acceleration voltages of 200 keV and 300 keV. The calculations for the fields of the electromagnetic mode, and the normalization terms were varified using COMSOL Multiphysics R ⃝, shown on figure S.4. 29 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Fiber diameter 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 50 100 150 200 250 300 Coherence length, Lc, [ m] Electron-fiber-coupling, for =1064 nm 0.18 0.185 0.19 0.195 0.2 0.205 0.21 0.215 0.22", + "R ⃝, shown on figure S.4. 29 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Fiber diameter 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 50 100 150 200 250 300 Coherence length, Lc, [ m] Electron-fiber-coupling, for =1064 nm 0.18 0.185 0.19 0.195 0.2 0.205 0.21 0.215 0.22 0.225 Fiber diameter 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 50 100 150 200 250 300 Coherence length, Lc, [ m] Electron-fiber-coupling, for =1064 nm Figure S.3: Relevant properties for the coupling of relativistic electrons to Si3N4-core (left) and Si-core (right) in vacuum. The waveguide width is chosen based on the optimal phase matching condition - note the divergence of Lc for 200 keV electrons at a diameter of 463 nm and 213 nm in Si3N4 and Si, respectively. The energy selectivity for these parameters is shown in figure 4b in the main text. This regime determines the coupling constant, and the characteristic decay length of the field out of the fiber (e−1 distance). Different diameters are optimized for either slower or faster electrons. 30 Figure S.4: (top) Distribution of the field component Ez in a step index fiber of a Si3N4 in vacuum. The inner circle is the core, and the outer two circles form the boundaries for the numerical box. (bottom) a cross-section of the above colormap. The field component Ez is normalized to represent the field per photon, in a fiber of length L = 100µm with periodic boundary conditions, multiplied by q 2ℏω0 L. That is, it is the coupling constant α for a electron that travels in a fixed distance parallel to such a fiber. The calculation is done for vacuum wavelength λ0 = 1064nm. The simulations verify the analytic calculation of the mode properties, used to evaluate the coupling constant. 31 Appendix A Assisting derivations • Explicit derivation of BCH According to BCH, for operators that obey [X, Y ] = const eX+Y = eXeY e−1 2 [X,Y ] (122) eX+Y = eY eXe+ 1 2 [X,Y ], (123) where I just stressed the importance of the sign. For the displacement operator, X = αˆba†, Y = −α∗ˆb†a, 1 2[X, Y ] = 1 2 h αˆba†, −α∗ˆb†a i = 1 2 |α|2 (124) . (125) which means that D(ˆbα) = eαˆba†−α∗ˆb†a (126) = e−1 2 |α|eαˆba†e−α∗ˆb†a (127) = e+ 1 2 |α|e−α∗ˆb†aeαˆba†. (128) • Approximating (N + ℓ)! ≈N! (N)ℓusing Stirling’s Formula, z! ≈ √ 2πz \u0000 z e \u0001z. Taking the natural logarithm result in ln (z!) ≈zln (z) −z + 1 2ln (2πz) . I will need to convert ln (N + ℓ) to a form with ln (N), so explicitly ln (N + ℓ) = ln \u0012 N \u0012 1 + ℓ N \u0013\u0013 (129) = ln (N) + ln \u0012 1 + ℓ N \u0013 , (130) which by Taylor expantion to the first order ,provides ln (N + ℓ) = ln (N) + ℓ N + O \u0012 ℓ N \u00132 . (131) Expanding the factorial (N + ℓ)! according to the above, ln ((N", + "(N) + ln \u0012 1 + ℓ N \u0013 , (130) which by Taylor expantion to the first order ,provides ln (N + ℓ) = ln (N) + ℓ N + O \u0012 ℓ N \u00132 . (131) Expanding the factorial (N + ℓ)! according to the above, ln ((N + ℓ)!) ≈(N + ℓ) ln (N + ℓ) −(N + ℓ) + 1 2ln (2π (N + ℓ)) (132) eq. (131) ≈ Nln (N) −N + 1 2ln (2πN) | {z } ln(N!) + (N + ℓ) ℓ N | {z } ℓ+ ℓ2 N +ℓ· ln (N + ℓ) | {z } ln(N)+ ℓ N −ℓ. (133) Neglecting terms of order \u0000 ℓ N \u00012, this resuls in ln ((N + ℓ)!) = ln (N!) + ℓln (N) + 2ℓ2 N . Within a correction of e( ℓ N ) 2 , one gets, (N + ℓ)! = N! (N)ℓe2 ℓ2 N + O \u0012 ℓ N \u00132 . For simplification, in the approximation of 2ℓ2 ≪N, this leads to the final result (N + ℓ)! ≈N! (N)ℓ. (134) Which we use to simplify the factorial terms in the electron-photon coupling and electron- electron coupling. 32 References [1] Armin Feist, Katharina E. Echternkamp, Jakob Schauss, Sergey V. Yalunin, Sascha Sch¨afer, and Claus Ropers. Quantum coherent optical phase modulation in an ultrafast transmission electron microscope. Nature, 521(7551):200–203, May 2015. [2] Marlan O. Scully and M. Suhail Zubairy. Quantum Optics. Cambridge University Press, Septem- ber 1997. [3] Leonard Mandel and Emil Wolf. Optical Coherence and Quantum Optics. Cambridge University Press, September 1995. [4] F. Javier Garc´ıa de Abajo. Multiple excitation of confined graphene plasmons by single free electrons. ACS Nano, 7(12):11409–11419, December 2013. [5] Katharina E. Echternkamp, Armin Feist, Sascha Sch¨afer, and Claus Ropers. Ramsey-type phase control of free-electron beams. Nature Physics, 12(11):1000–1004, November 2016. [6] C. Yeh. Guided-wave modes in cylindrical optical fibers. IEEE Transactions on Education, E-30(1):43–51, February 1987. [7] Chin-Lin Chen. Foundations for Guided-Wave Optics. John Wiley & Sons, September 2006. Google-Books-ID: LxzWPskhns0C. 33", + "1 The general traits of inelastic electron scattering by the adsorbed system A.R. Cholach∗ and V.M. Tapilin Boreskov Institute of Catalysis, Prospekt Akademika Lavrentieva, 5 630090 Novosibirsk, Russian Federation Abstract Inelastic electron scattering by the adsorbate covered Pt(100) single crystal surface is studied by Disappearance Potential Spectroscopy and density of states (DOS) calculations. Two peculiar channels of elastic electron consumption are highlighted, both related to the substrate core level excitation coupled separately with two particular electron transitions. The first channel affects the adsorbed layer and enables to reveal the valence state structure of the adsorbed species as well as the substrate DOS. The second one includes the multiple plasmon oscillations. The proposed mechanism of electron transitions assumes that one-dimensional DOS at the vacuum level is an additional spot for location of excited electrons, along with vacant DOS at the Fermi level. Observed phenomena are supposed to be a general regularity of electron-solid interaction and a useful tool for fingerprinting the adsorbed layer at molecular level. Keywords: inelastic electron scattering; coupled electron excitation; DAPS; vacant state structure; plasmon oscillations ∗Corresponding author. Tel.: +7 383 326 95 40; fax: +7 383 330 80 56; E-mail address: cholach@catalysis.ru 2 1. Introduction Disappearance Potential Spectroscopy (DAPS) is based on the threshold core level excitation of target atoms by the primary electron beam of time-based energy Ep [1]. As a result of energy loss, a part of incident electrons disappears from elastic current I(Ep) and provides a sharp drop in the dI(Ep)/dEp = f(Ep) dependence being recorded experimentally. In the course of excitation operative, the core and primary electrons occupy available vacant states just above the Fermi level EF, and so the self-convolution of vacant states density is in charge of the spectral structure. The spectral feature is accordingly localized at the energy point equal to difference between the core level and the vacant state energy. The high surface sensitivity ~1-3 monolayer defined by half the electron mean free path in a solid makes DAPS a promising tool in adsorption and heterogeneous catalysis investigations. However, this technique is not widely adopted so far mainly due to the following reasons. According to DAPS principle the spectral structure depends not only on a surface content, but greatly on a given density of vacant states structure, and so DAPS looses as analytical techniques in comparison with Auger-electron (AES) and X-ray Photoelectron Spectroscopy (XPS). The peculiarities of the vacant state structure is generally the main outcome of conventional DAPS application [1]. Furthermore, an appreciable density of vacant states at EF is a precondition for obtaining the clearly defined spectrum. It is not the case for nearly filled outer shell of the Platinum group metals which are of great practical importance as active components of a good deal of industrial catalysts [2]. In particular, there is the only paper with a very few comments related to the Pt3d5/2 DAPS study by the primary electron beam of above 2 keV energy [3]. We have established earlier a perfect agreement between calculated and experimental DAPS spectra around the Pt4d5/2 core level obtained for the", + "In particular, there is the only paper with a very few comments related to the Pt3d5/2 DAPS study by the primary electron beam of above 2 keV energy [3]. We have established earlier a perfect agreement between calculated and experimental DAPS spectra around the Pt4d5/2 core level obtained for the hydrogen and oxygen covered Pt(100)-(1x1) single crystal surface [4], but this result is mainly of theoretical importance since the valence rather than the vacant states are responsible for adsorption, catalytic and other special properties of a solid. A shake-off process is quite a common occurrence that is often observed by means of AES and XPS [5,6]. A set of regular spectral features similar to shake-off satellites of isolated molecules [7] was revealed in Pt4d5/2 DAPS spectra after formation of different adsorbed layers on the Pt(100)-(1x1) single crystal surface [8]. The number and locations of these features above the threshold are well correlated with number and ionization potentials of corresponding valence states of the adsorbed particles determined for analogous systems by Ultraviolet Photoelectron Spectroscopy (UPS). Thus a single H1s or O2p satellite was detected after H2 or O2 adsorption; three satellites related to ionization of 1π, 5σ, 4σ and 2π, 1π+5σ, 4σ states were revealed after CO and NO adsorption, respectively. This phenomenon was regarded as an additional channel of 3 inelastic electron scattering, namely the coupled (conjugate) electron excitation (CEE) which includes the threshold core level excitation of substrate atom and ionization of the valence state of the adsorbed species. Besides that, a multiple bulk and surface Pt plasmon excitations are believed to be also coupled with the core level excitation [9]. The previous paper [8] is just a sequence of very promising results needed to be confirmed and clearly understood. This work highlights the novel kind of electron transition affecting an occupied, substrate and adsorbate density of states (DOS) which has never been considered in a standard theory of DAPS. The proposed mechanism of such a transition assumes a pronounced DOS at the vacuum level to be an additional spot for location of excited electrons, along with the vacant DOS at EF. The present work includes reinterpretation of fine DAPS spectral structure in light of an advanced DOS calculations, partially reiterates previous data using an improved spectra processing, and shows new experimental results related to H2 adsorption. The paper as a whole reveals the advanced possibilities of simple and accessible DAPS technique for investigation the adsorptive, catalytic and other special properties of solids at the molecular level. 2. Experimental and theoretical Experiments were performed using the ultrahigh vacuum chamber with residual gas pressure of < 2·10-8 Pa. The chamber was equipped with low-energy electron diffraction (LEED), AES with retarding field analyzer, monopole mass-spectrometer and Ar+ ion gun. The DAPS technique was realized on the basis of 3-grid LEED optics. The central LEED electron gun with a tungsten filament was used to form the primary electron beam of ~ 1 mm size, ~ 1 μA current with normal incidence to the sample surface. The time-based accelerating potential was applied to W-filament. The", + "realized on the basis of 3-grid LEED optics. The central LEED electron gun with a tungsten filament was used to form the primary electron beam of ~ 1 mm size, ~ 1 μA current with normal incidence to the sample surface. The time-based accelerating potential was applied to W-filament. The outside LEED grids were grounded while the middle one was at the same negative potential as the electron gun filament, so that only quasi-elastic electrons scattered by the target could pass it to reach the LEED screen as a collector. The pass band cutoff 2-3 eV of the retarding grid energy analyzer was kept constant over a given set of experiments. The spectra were recorded as a first derivative of total collector current by the modulation of sample potential (~ 0.5 Vp-p; ~ 2 kHz). The average spectral resolution was ~ 1.0 eV, the peak position could be determined within ~ 0.1 eV precision. The Pt(100) single crystal of 99.999% purity, ~ 1cm width and ~ 1 mm thickness was oriented within ~ 1º. The thermodynamically favorable Pt(100)-hex surface structure is known to reconstruct readily into (1x1) one under adsorption of certain molecules [10]. In order to keep stable the substrate structure all adsorption experiments were performed at room temperature on the AES-clean (1x1) surface which was obtained according to “NO-receipt” [11]. 4 The DAPS spectra methodically accumulate an attendant diffraction features which are up to 103 times more intensive than the true DAPS peaks for well-ordered single crystal surface. And so the difference spectra (the adsorbate covered surface minus the clean one) were considered where diffraction features suppress each other under subtraction. Of course, the Pt spectral contribution also becomes partially self-suppressed; therefore those features are revealed which were changed under adsorption taking part in the adsorbate-surface bond formation. The difference spectra for the CO/Pt(100) system were still complicated by the strong diffraction peaks originated from the well-ordered adsorbed layer. In order to eliminate this background a smooth Gaussian fit of each spectrum was subtracted from the difference one. Above procedures are commonly used in data processing to find out the fine spectral structure without disturbance [12]. The DAPS measurements were performed in the neighborhood of Pt4d core level excitation to minimize the diffraction background, a detailed description of sample treatment can be found elsewhere [2,8]. The reference EF point in all spectra corresponds to the incident electron energy of 314.8±0.5 eV that is close to reference value of the Pt4d5/2 core level energy 314.6 eV [12] with due regard for the filament work function [1,8]. The calculations of bulk Pt DOS were performed by the Quantum Espresso package [13] for experimental lattice parameters. Core electrons were treated with the Perdew-Burke-Ernzerhof ultrasoft pseudopotential [14], the plane waves with Ecut = 40 Ry were used for basis set of the valence electron wave function. A numerical integration in the reciprocal space was performed within a 12x12x12 grid. The present technique permits to calculate precisely the vacant state DOS and to remediate deficiencies of earlier calculations [8] suffered from the poor basis set.", + "Ry were used for basis set of the valence electron wave function. A numerical integration in the reciprocal space was performed within a 12x12x12 grid. The present technique permits to calculate precisely the vacant state DOS and to remediate deficiencies of earlier calculations [8] suffered from the poor basis set. The primary electron is described by the plane wave with vector component parallel to the surface. 0 || = k r 3. Results and discussion 3.1. The substrate DOS probing by DAPS The threshold Pt4d core level excitation is the main channel of the primary electron energy consumption traced by DAPS in which two electrons, the primary and the core level electron take part: − p e 2 2 ,1 9 2 / 5 0 2 ,1 10 0 4 4 a d a d ep + → + + − , (1) where a1,2 stands for the Pt vacant states. It is commonly accepted that no information on the occupied part of valence bands can be obtained from DAPS spectra. However, numerous regular CEE satellites gives evidence that the 5 main channel (1) is coupled with ionization of the adsorbate valence band; therefore a three- electron process can also join the primary electron energy consumption (1). In this process, in addition to two operating electrons excited to the vacant a-states just above EF, a valence electron at state V of the adsorbed species moves to vacuum level and thus leaves the system as . Therefore, the electron transitions (1) should be corrected as: − em e − − + + + → + + + em p e a V d a V d e 2 2 ,1 1 9 2 / 5 0 2 ,1 2 10 0 4 4 (2) In the general case of transition (2) three electrons occupy some vacant states with energies ε1, ε2 and ε3, and one of these electrons is excited at that from the valence state with energy εV. And so the probability W(E) of process (2) is determined by the DOS self-convolution: ∫ ∫ ∫ + − − − = E V E d d d E W 0 3 2 1 3 0 3 2 0 2 1 1 ) ( ) ( ) ( ) ( ) ( 2 1 ε ε ε ε σ ε σ ε ε σ ε ε σ ε ε ε , (3) where σ is the valence band DOS and E is energy excess above the threshold energy. Fig. 1 The DOS of bulk Platinum (black) and of free space (magenta) relative the Fermi level. 6 Strictly speaking one should consider the local DOS which defines the bulk-to-vacuum DOS transformation. For qualitative understanding of the spectral structure we have considered only extreme cases, namely the bulk Pt DOS and the vacuum space shown in Fig. 1. Fig. 2 The DAPS spectra obtained after various exposure of clean Pt(100) surface to CO; red curve: shake-off satellite structure expected from data of Fig. 1. The scattering of electron waves with wave vectors 1k r and 2k", + "the bulk Pt DOS and the vacuum space shown in Fig. 1. Fig. 2 The DAPS spectra obtained after various exposure of clean Pt(100) surface to CO; red curve: shake-off satellite structure expected from data of Fig. 1. The scattering of electron waves with wave vectors 1k r and 2k r in states with wave vectors q k r r ± 1 and q k r m r 2 is governed by the Fourier transform of Coulomb interaction 1/|q|2, therefore the scattering with small changes of wave vectors is preferable. It means that the number of vacant states to which the scattering is possible is proportional not to the total density of states, 7 but rather to its part near the initial wave vectors. That is the reason to use one-dimensional DOS ~ ) ( / 1 k E k r r ∇ for the vacant states where ) (k E r is the band energy, instead of the total DOS obtained by integration of ) ( / 1 k E k r r ∇ over the constant energy surface in the wave vector space. In the vacuum region for a free electron the total DOS ~ E1/2, while the 1-D DOS ~ E-1/2 [15] as shown in Fig. 1. The unoccupied part of the bulk Pt DOS is significant only right above the Fermi energy as in the case of well-ordered Pt(111) surface [16]. The DOS is small and nearly constant at larger energy, and so the main transition of core electron will occur only in the narrow energy region around EF. On the other hand, the vacuum states shown in Fig. 1 exhibit a strong maximum (infinite theoretically) at the work function of clean Pt (100)-(1x1) surface ϕ = 5.6 eV [17,18]. A generally accepted theory of the threshold core level excitation does not consider these states as a possible spot for location of excited electrons [1], but our previous experimental observations [8] reaffirmed by the present ones give evidence that it can be the case. Therefore the DOS above EF in Fig. 1 can be approximately represented by two discrete levels at and F E ϕ + F E E W ( that permits to reduce expression (3) to the following one: ϕ ϕ σ ϕ σ σ σ σ > − + − ≈ E F F F E E E E E ) ( ) ( ) ( ( ) ( ) 2 3 (4) > + E) 0 Fig. 2 shows the difference DAPS spectra obtained after various exposure of clean Pt(100) surface to CO. In addition to expected 1π, 5σ, 4σ CEE satellites there is a pronounced spectral structure within the range of 8-12 eV. It should be noted that numerous theoretical calculations and UPS measurements definitely relate the similar spectral features to Pt DOS [19-21]. If the Pt DOS also takes place in data of Fig. 2 then outgoing electron should overcome a sum of the valance state energy relative EF and the work function barrier. Thus at the total energy point one should", + "measurements definitely relate the similar spectral features to Pt DOS [19-21]. If the Pt DOS also takes place in data of Fig. 2 then outgoing electron should overcome a sum of the valance state energy relative EF and the work function barrier. Thus at the total energy point one should observe a spectral feature in experimental dI(Ep)/dEp dependence corresponding to the shake-off satellite. The red curve in Fig. 2 was constructed from data of Fig. 1 in the following way. Accounting for aforementioned details of spectra recording, the Pt DOS below EF was reversed, differentiated and shifted to the right by the Pt work function according to the second term in relation (4). Therefore, the red curve simulates the expected spectral structure assuming a direct electron transition from the Pt valence band to a narrow localized state at the vacuum level. Fig. 2 displays a perfect correspondence between simulated and experimental spectral features in the range of 8-12 eV. 8 Fig. 3 The DAPS spectra obtained after shown exposure of clean Pt(100) surface to a given adsorbate; red curve: shake-off satellite structure expected from data of Fig. 1. Fig. 3 exhibits a set of referred CEE satellites in DAPS spectra related to different adsorbed layers. In addition, the similar structure at the 8-12 eV energy range is clearly seen in each spectrum as well as in the red curve copied from Fig. 2. Therefore Fig. 2 and Fig. 3 give a direct evidence for that the valence state V in equation (2) may belong to the substrate atom as well as to the adsorbed particle. 3.2. The generality of CEE effect Fig. 4 shows a set of raw analogous DAPS spectra obtained after exposure of clean Pt(100) surface to O2. The relevant spectral features are summarized in Table 1. The Pt DOS in Fig. 1 9 showed no fine vacant states structure close to the Fermi level. In order to assign a set of distinct peaks at 2-7 eV in Fig. 4 we assume that a narrow vacant state just above EF alike that at the vacuum level can be also a spot for shake-up transitions of the Pt valence band electrons. The blue curve in Fig. 4 simulates this kind of electron excitation in the same way as above, but without the work function shift according to the first term in relation (4). An acceptable correspondence between experimental and simulated spectral structure testifies the possibility of such a transition. Fig. 4 The raw difference DAPS spectra obtained after indicated exposures of clean Pt(100) surface to O2; blue curve: shake-up satellite structure expected from data of Fig. 1. 10 Table 1 Relevant spectral features in the extended DAPS spectra of Figs. 2-5. E-EF, eV Peak assignment Comment 0 Pt4d5/2 The reference point (Fermi level) 2-7 shake-up Occupied Pt5d-band satellites; Figs. 4, 5 8-12 shake-off Occupied Pt5d-band satellites; Figs. 2, 3, 5 17.2 Pt4d3/2 Pt4d3/2 core level (Pt4d spin-orbit splitting); Fig. 4 14.0 O2p CEE of the Oad species, relative Pt4d5/2; Fig. 4 31.5 O2p CEE of the Oad species, relative Pt4d3/2; Fig.", + "(Fermi level) 2-7 shake-up Occupied Pt5d-band satellites; Figs. 4, 5 8-12 shake-off Occupied Pt5d-band satellites; Figs. 2, 3, 5 17.2 Pt4d3/2 Pt4d3/2 core level (Pt4d spin-orbit splitting); Fig. 4 14.0 O2p CEE of the Oad species, relative Pt4d5/2; Fig. 4 31.5 O2p CEE of the Oad species, relative Pt4d3/2; Fig. 4 9.2 ħωs Relative Pt4d5/2; Fig. 4 11.8 ħωb Relative Pt4d5/2; Fig. 4 21.2 ħωs+ħωb Relative Pt4d5/2; Fig. 4 24.2 2ħωb Relative Pt4d5/2; Fig. 4 36.0 3ħωb Relative Pt4d5/2; Fig. 4 A pronounced peak in Fig. 4 at 14.0 eV stands for the CEE satellite related to ionization of the O2p state [22] coupled with the Pt4d5/2 threshold excitation core level as discussed above [8]. Furthermore, a splitting of this feature at top exposure may be considered as the formation of new adsorbed state of oxygen atoms Oad, e.g. like the bridge state or incorporated one [23,24]. In this way the peak at 31.5 eV is also O2p satellite, but coupled now with the Pt4d3/2 core level excitation. As expected, the intensity of both O2p satellites increases on coverage and they are localized apart each other at characteristic spin-orbit Pt4d splitting [25]. According to electron theory of metals every core level can undergo a threshold excitation, and there are no reasons prohibiting the CEE process related to excitation of the other core level than the Pt4d one. The present observation in Fig. 4 corroborates this statement and implies that the other satellites alike O2p one, and therefore the valence state structure of the adsorbed species will be coupled with the threshold excitation of any substrate core level. The multiplicity and regularity of similar spectral satellites revealed from different adsorbed layers suggests that the CEE effect is believed to be a fundamental regularity of inelastic electron scattering in any adsorbed system. In theory, CEE monitoring enables to determine a localization of the adsorbed species at multicomponent surface, because CEE satellites should accompany threshold excitation of only that surface atom which is chemically bound to a given particle. In turn, the reference core level energies of different elements are easily distinguishable. It should be noted that it is a high surface sensitivity of DAPS technique that provides a substantial spectral contribution of CEE satellites originated from the adsorbed layer. 11 3.3. Plasmon excitation as another kind of the conjugate electron excitation Plasmon excitation is an effective channel of inelastic electron scattering. It is responsible for the most intensive peaks displayed for different surfaces by (Reflection) Electron Energy Loss Spectroscopy (R)EELS [26]. For instance, EELS spectra obtained for the clean Mo(110) [27], W(111), (112) [28] and Si(100) [29] single crystal surfaces reveal a set of 10-20 features with the base width ~ 5 eV related to multiple surface and bulk plasmon excitations. Interband transitions make difficult an identification of plasmon oscillations in EELS spectra. A transition with energy E is usually assigned as a bulk plasmon excitation if multiple transitions divisible by E and the surface plasmon transition 2 / E are also revealed [30]. A set of not intensive, but systematic narrow peaks are", + "difficult an identification of plasmon oscillations in EELS spectra. A transition with energy E is usually assigned as a bulk plasmon excitation if multiple transitions divisible by E and the surface plasmon transition 2 / E are also revealed [30]. A set of not intensive, but systematic narrow peaks are perceptible in spectra of Fig. 4 and listed in Table 1. The general regularities of these features consist in the following. First, in contrast to O2p satellites the peak intensity decreases on oxygen coverage. Second, three of them are localized at invariable distance of 12.0±0.2 eV apart each other including the first satellite relative EF. This behaviour is typical for the multiple bulk plasmon excitations resulted in the energy loss aliquot to plasmon frequency ħωb [27-29]. Then the feature at 9.2 eV corresponds to surface plasmon ħωs with an expected ratio of ωb/ωs ~ 2 [31]. The peak at 21.2 eV is similarly assigned to the multiple bulk and surface plasmon excitation of the total ħωs+ħωb energy. The weak features in the range of 26-30 eV can be attributed to the coupled plasmon and Pt4d3/2 core level excitation. It is important to note that the reference point of plasmon locations is the substrate Fermi energy which corresponds to threshold excitation of the Pt4d5/2 core level. In other words, the coupled plasmon and core level excitation can be considered as another kind of CEE phenomenon discussed above. Fig. 5 shows the difference DAPS spectra obtained after exposure of clean Pt(100) surface to H2. The above consideration enables to assign properly all relevant features. The H1s satellite is the same as in Fig. 3. The shake-off emission from the Pt valence band is responsible for the spectral structure at 8-12 eV similar to Fig. 2 and Fig. 3. The shake-up transition of the Pt valence band stands for the spectral features at 2-7 eV as in Fig. 4. The peak at ~ 13.3 eV probably results from the bulk plasmon excitation ħωb, and its intensity decreases on Had exposure as expected. Then the peak at ~ 10.0 eV at 0.5 L H2 exposure corresponds to surface plasmon ħωs, and it disappears at higher H2 exposure. The decrease in plasmon features intensity on coverage in Fig. 4 and Fig. 5 is certainly due to the shielding effect of the Oad and Had layer, respectively. 12 Fig. 5 The DAPS spectra obtained after indicated exposure of clean Pt(100) surface to H2; blue and red curve: shake-up and shake-off satellite structure expected from data of Fig. 1, respectively. The reliability of plasmon energy can be evaluated from Table 2. The present plasmon energy has to be refined in the following way. First, it should be shifted up by half the typical base width 1.5-2.5 eV of REELS peak for the proper comparison with reference data, because Fig. 4 and Fig. 5 represent DAPS spectra in a derivative mode. On the other hand, it should be shifted down by the double energy of vacant a-state ~ 1 eV where two operating electrons are finally localized according to the", + "peak for the proper comparison with reference data, because Fig. 4 and Fig. 5 represent DAPS spectra in a derivative mode. On the other hand, it should be shifted down by the double energy of vacant a-state ~ 1 eV where two operating electrons are finally localized according to the prime process (1). Taking into account these corrections the agreement between the reference data and the present value of ħωb = 12.0÷13.3 eV seems satisfactory. Theoretical estimation of the bulk plasmon excitation energy is performed by the free electron gas formula [30,37]: 13 2 / 1 2 4 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = m ne b π ω , (5) where n and m stands for effective number of free electrons per Pt atom (supposed to be s and p electrons) and effective electron mass, respectively. Table 2 The energy of surface ħωs and bulk ħωb plasmon excitation, and characteristic peak base width for different surfaces, eV Surface Peak base width ħωs ħωb Comment Al-film ~ 3 10.3 15.3 EELS [32] Al-film ~ 5 10 15 REELS [33] Al(100) ~ 5 10.5 14.9 REELS [31] Pt-foil ~ 3 15.2 REELS [34] Pt(100) ~ 5 13.9 13.5 REELS [35] Theory [36] O/Pt(100) H/Pt(100) ~ 1 9.2 10.0 <12.0> 13.3 DAPS, Fig. 4 DAPS, Fig. 5 Pt 2 free electrons 3 free electrons 9.1 11.2 12.9 15.8 Formula (5) Experimental plasmon energy is usually lower than followed from the free electron gas model [38]. Table 2 indicates the number of free electrons between 2 and 3 per Pt atom. The presently calculated electron configuration in Platinum crystal 5d86s2 supports this statement if d-electrons are treated as localized. Plasmon oscillations can be detected by means of AES and XPS [31]. In both cases a part of secondary electron energy aliquot to linear combination of ħωs and ħωb is consumed for plasmon excitations. Therefore spectral satellites are observed at corresponding positions at the lower- energy side of the main KLL Al(111), KLL Si(100) [39,40] peak or along the higher-energy side of the main C1s [40,41], Ge2s [42], Al2s and Al2p [33] peak. The physics and surface sensitivity of plasmon losses in XPS and EELS are quite similar, except the presence of electron–hole interaction and the lack of collimated beam in the former case which is qualified as a core-level loss spectroscopy [43]. In both cases plasmon oscillations come true due to energy loss by the primary- or photo-electron of arbitrary energy, and so the plasmon spectral features can be considered as shake-up satellites. In contrast to that, plasmon oscillations listed in Table 1 can be observed conditionally: in the presence of the core level excitation as a basic process, and if there 14 is an excess of primary electron energy above the Pt4d threshold. That is why plasmon features are localized at higher energy region with respect to the Fermi level. Furthermore, the scattered electron providing plasmon oscillations in EELS, AES and XPS measurements is actually an external one. On the contrary, an electron participating in CEE process can be duly regarded as internal", + "That is why plasmon features are localized at higher energy region with respect to the Fermi level. Furthermore, the scattered electron providing plasmon oscillations in EELS, AES and XPS measurements is actually an external one. On the contrary, an electron participating in CEE process can be duly regarded as internal one since it has just taken part in the core level excitation and became indistinguishable among other electrons within the integrated, substrate and adosbate DOS. The difference in excitation mechanism enhances an efficiency of electron energy dissipation and results in a relatively high resolution of plasmon satellites in DAPS spectra. Thus, the typical peak base width in CEE processes is ~ 1 eV (Fig. 4, Fig. 5), whereas that of plasmon features observed by EELS and XPS is 5-10 eV and 20-30 eV, respectively [39-42]. According to Fig. 1, the low density of vacant states close to EF necessary for the leading process of the Pt4d5/2 core level excitation results in low intensity of plasmon satellites in present DAPS spectra. The same reason is mainly responsible for the shortage of true DAPS peaks. Indeed, only a small shoulder in the range of 0 - 1 eV can be related to the prime process (1) in Fig. 4 and Fig. 5. Of course, the aforementioned procedure of spectra subtraction also self-suppresses this spectral contribution. 4. Conclusion Fig. 6 displays a general picture of specific electron transitions coupled with the basic process of the threshold core level excitation in any adsorbed system. Fig. 6 The general picture of electron transitions, as an example, in the H/Pt(100) adsorbed system exposed to the primary electron beam of above E(Pt4d5/2) core level energy. 15 These transitions are independent channels of the primary electron energy consumption and can be classified as follows. (1) A sequential shake-off electron transition from the valence states of the adsorbed species to vacuum level (indicated in Fig. 6 with a red arrow related to H1s state). The probability of this process is represented by the second term of relation (4). According to energy conservation the spectral feature is then observed at the incident electron energy ϕ ε + + = V d p Pt E E ) ( 2 / 5 4 , where is a discrete core level energy; ) ( 2 / 5 4d Pt E ϕ is a work function; V ε is a valence state energy of the adsorbed species. Therefore this channel provides a direct experimental data on the density of valence states ) (ε σV of the adsorbed layer as a single or a set of CEE satellites above EF as takes place in Figs. 2, 3. (2) In the same terms the probability of shake-off electron transition from the occupied part of Pt 5d-band to vacuum level is also represented by the second part of relation (4) (indicated in Fig. 6 with a red arrow related to Pt DOS). The corresponding spectral satellites then appear in the energy range of ϕ + |) ( | PtDOS E and reveal the Pt DOS structure relative the reference", + "level is also represented by the second part of relation (4) (indicated in Fig. 6 with a red arrow related to Pt DOS). The corresponding spectral satellites then appear in the energy range of ϕ + |) ( | PtDOS E and reveal the Pt DOS structure relative the reference point EF +ϕ as in Figs. 2, 3, 5. Note that electron transitions (1) and (2) revealed by DAPS are similar to those in a well-proven UPS technique. The only difference concerns the energy source – an overexcited core electron and photon, respectively. (3) The probability of shake-up electron transition from the Pt valence band to vacant state at EF is represented by the first part of relation (4) and indicated in Fig. 6 with a blue arrow. Similarly to item (2) the spectral structure reproduces the Pt DOS right above EF, but without the work function shift, as takes place in Figs. 4, 5. Fig. 6 displays the transitions between one-electron states and should be supplemented with the collective electron shake-up transitions related to multiple plasmon excitations, which can be also regarded as another kind of the conjugate electron excitation. To recapitulate, inelastic electron scattering by the adsorbed system is characterized by two peculiar channels related to specific coupled electron excitations. 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Sci. 255 (2009) 6598.", + "Skip to main content [image] We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate [image] Help | Advanced Search All fields Title Author Abstract Comments Journal reference ACM classification MSC classification Report number arXiv identifier DOI ORCID arXiv author ID Help pages Full text Search Login This version of 1602.07484v2 has been withdrawn and is unavailable See abstract page for more details • About • Help • Contact • Subscribe • Copyright • Privacy Policy • Web Accessibility Assistance • arXiv Operational Status Get status notifications via email or slack", + "Electron bunch generation from a plasma photocathode A. Deng1*, O. Karger2*, T. Heinemann2,3,4,5, A. Knetsch5, P. Scherkl3,4, G. G. Manahan3,4, A. Beaton3,4, D. Ullmann3,4, G. Wittig2, A. F. Habib3,4, Y. Xi1, M. D. Litos6, B. D. O’Shea7, S. Gessner7, C. I. Clarke7, S. Z. Green7, C. A. Lindstrøm8, E. Adli8, R. Zgadzaj9, M. Downer9, G. Andonian1,10, A. Murokh10, D. L. Bruhwiler11, J. R. Cary12, M. J. Hogan7, V. Yakimenko7, J. B. Rosenzweig1, B. Hidding3,4 1 Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095, USA. 2 Department of Experimental Physics, University of Hamburg, Hamburg 22761, Germany. 3 Scottish Universities Physics Alliance, Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK. 4 Cockcroft Institute, Sci-Tech Daresbury, Keckwick Lane, Daresbury, Cheshire WA4 4AD, UK. 5 Deutsches Elektronen-Synchrotron DESY, Hamburg 22607, Germany. 6 Center for Integrated Plasma Studies, University of Colorado Boulder, Boulder, Colorado 80303, USA. 7 SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA. 8 Department of Physics, University of Oslo, Oslo 0316, Norway. 9 Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA. 10 RadiaBeam Technologies, Santa Monica California 90404, USA 11 RadiaSoft LLC, Boulder, Colorado 80304, USA 12 Tech-X Corporation, Boulder, Colorado 80303, USA * These authors contributed equally to this work. Plasma waves generated in the wake of intense, relativistic laser1,2 or particle beams3,4 can accelerate electron bunches to giga-electronvolt (GeV) energies in centimetre-scale distances. This allows the realization of compact accelerators having emerging applications, ranging from modern light sources such as the free-electron laser (FEL) to energy frontier lepton colliders. In a plasma wakefield accelerator, such multi-gigavolt- per-metre (GV m-1) wakefields can accelerate witness electron bunches that are either externally injected5,6 or captured from the background plasma7,8. Here we demonstrate optically triggered injection9,10,11 and acceleration of electron bunches, generated in a multi-component hydrogen and helium plasma employing a spatially aligned and synchronized laser pulse. This “plasma photocathode” decouples injection from wake excitation by liberating tunnel-ionized helium electrons directly inside the plasma cavity, where these cold electrons are then rapidly boosted to relativistic velocities. The injection regime can be accessed via optical11 density down-ramp injection18,19,20, is highly tunable and paves the way to generation of electron beams with unprecedented low transverse emittance, high current and 6D-brightness12. This experimental path opens numerous prospects for transformative plasma wakefield accelerator applications based on ultra- high brightness beams. The advent of photoinjectors in state-of-the-art linear accelerators (linacs) has enabled the substantial increases in electron beam quality that have ushered in an era of new scientific capabilities, as exemplified by the introduction of the hard X-ray FEL13. These photoinjectors produce electron beams in electric fields of ~100 megavolts-per-metre (MV m-1). This injection environment largely determines key beam qualities such as the transverse emittance (phase space area) and beam brightness. The strong accelerating field restricts emittance dilution and pulse lengthening by quickly increasing the relativistic Lorentz factor of the beam, γ = (1-v2/c2)- 1/2 , where v is the electron velocity and c is the speed of light, thus diminishing these effects that arise from space charge", + "and beam brightness. The strong accelerating field restricts emittance dilution and pulse lengthening by quickly increasing the relativistic Lorentz factor of the beam, γ = (1-v2/c2)- 1/2 , where v is the electron velocity and c is the speed of light, thus diminishing these effects that arise from space charge forces14 and scale as γ-2. As the electric fields in a plasma wakefield accelerator (PWFA) can exceed those of photoinjectors by more than two orders of magnitude, the PWFA is an attractive environment for high brightness electron beam creation. A scenario termed plasma photocathode aims to liberate electrons directly inside such plasma wakes via tunneling ionization of neutral gas species at the tight focus of a dedicated, low- power laser pulse. These electrons therefore have low transverse momentum and small initial phase space volume, which is why this injection, which is decoupled from the rapid acceleration process per se, permits beams with dramatically improved quality. The process is freely tunable by adjusting injection laser parameters and neutral gas density independent of the acceleration process. It allows transfer of energy from a relatively low-quality driver electron beam into an injected witness bunch of much higher quality. In this way it offers a path to ultrahigh brightness, exceeding the state-of-the-art by many orders of magnitude. We unlocked the plasma photocathode regime by discovery of a transition process from optical plasma density down-ramp injection11 in an electron beam-driven plasma wave. This was realized at the Facility for Advanced Accelerator Experimental Tests (FACET) at the SLAC National Accelerator Laboratory15. A pre-ionized plasma channel is formed in a hydrogen- helium gas mixture by focusing an ~800 nm Ti:sapphire laser pulse with an axilens16,17 (see Extended Data Figure 1). The resulting laser intensity distribution with peak intensity of Ipre ~ 3  1014 W cm-2 selectively tunnel-ionizes hydrogen but not helium, generating a tens-of- centimetre long hydrogen plasma channel of electron density ne,H ≈ 1.3  1017 cm-3 with limited varying width up to 130 µm. This channel accommodates a dark-current-free21, multi-GV m-1 wakefield, driven by intense electron drive beams with root-mean-square (r.m.s.) length of z  30 µm (see Methods). A second, perpendicular-oriented laser pulse is focused to Iinj ~ 1015 W cm-2 by an off-axis parabolic mirror to release helium electrons from the dual-component gas medium to achieve injection. If this laser pulse crosses the electron driver propagation axis before the electron beam, the plasma wave encounters a plasma density spike due to the presence of the additional localized helium plasma. If this density spike is pronounced enough, it distorts the plasma wave substantially, leading to the injection of hydrogen and helium electrons originating from outside the plasma wave as shown in Figure 1a-c. This is optically- triggered down-ramp injection18,19,20, which we term plasma torch injection. In contrast, if the laser pulse arrives at exactly the right time – slightly after the driver beam – it releases helium electrons directly within the plasma wave, as shown in Figure 1d-f. This is the plasma photocathode injection mode, which does not require plasma wave perturbation, and in fact", + "In contrast, if the laser pulse arrives at exactly the right time – slightly after the driver beam – it releases helium electrons directly within the plasma wave, as shown in Figure 1d-f. This is the plasma photocathode injection mode, which does not require plasma wave perturbation, and in fact profits from lower laser intensities as these produce colder electron populations – direct beam implementation is the only change introduced by the laser in this case. Figure 1 | Three-dimensional particle-in-cell (PIC) simulations of two injection modes in the beam-driven plasma wave. a-c, sequential snapshots of the plasma torch injection process, where the driver beam electrons (blue dots) move to the right in the co-moving frame (=z-ct) and interact with pre-ionized hydrogen plasma channel electrons (colour-coded energy) and a perpendicular plasma filament of ionized hydrogen and helium. This plasma density spike, generated by a 5 mJ laser pulse, distorts the nonlinear plasma “blowout” shape and triggers injection of electrons from outside the blowout, as indicated by selected trajectories (green lines). d-f, sequential snapshots of the plasma photocathode injection process. A 0.5 mJ laser pulse (red) releases helium electrons via tunneling ionization inside the hydrogen-based plasma wave. Selected electron trajectories (green lines) show that the injected electrons originate from inside the plasma blowout. Snapshots c and f are taken when the respective witness bunches are fully formed, respectively. =0 is defined by the centre of the electron driver beam in the co-moving frame. Only particles within the central slice -2.5 µm < x < 2.5 µm are shown. Experimentally, an independently controllable fraction of the laser, having pulse duration of 65 fs full-width at half maximum (FWHM) and tunable energy, was focused to a spot size of w0  20 µm. Focal intensities of Iinj > 1015 W cm-2 are achieved with millijoule (mJ) energies and ionize a narrow filament of helium that intersects the driver beam propagation axis at 90° (see Methods). Plasma torch injection is relatively insensitive to electron-beam-to-laser timing, and is achieved for a sufficiently dense plasma filament as long as it spatially overlaps with the driver electron beam and is generated a few femtoseconds up to hundreds of picoseconds or more prior to driver beam arrival. Density down-ramp injection is a promising injection method in its own right but it had not yet been accomplished in beam-driven plasma accelerators before. Here we found it in a straightforward manner due to the steep density gradient19 obtainable by optically generated down-ramps. Figure 2 demonstrates laser-triggered plasma torch injection by switching the laser on and off repeatedly at 5 mJ over a consecutive series of shots. Experimental evidence of injected electrons is recorded in two ways: as excess electron charge downstream of the plasma source and by the appearance of an energetically distinct grouping of electrons at the spectrometer (see Methods). Figure 2 | Electron charge and spectra obtained from the plasma torch injection process over 360 consecutive shots. a, injected witness beam charge measured as charge difference at beam position monitors before and after the plasma source. The injection laser pulse", + "distinct grouping of electrons at the spectrometer (see Methods). Figure 2 | Electron charge and spectra obtained from the plasma torch injection process over 360 consecutive shots. a, injected witness beam charge measured as charge difference at beam position monitors before and after the plasma source. The injection laser pulse arriving 1 ps ahead of the electron driver beam is switched on and off repeatedly every 60 shots, as indicated by the red trace. b, corresponding electron spectra measured on the downstream magnetic imaging spectrometer, with optics set to image low energy electrons. The features at the top of the spectra are the decelerated driver beam electrons. With spatial alignment established via observation of the plasma torch injection, a timing delay scan was performed to determine when the injection laser arrives too late to generate the helium plasma filament prior to arrival of the plasma wave. Beyond this time, plasma torch injection is no longer accessible, as verified by the electron charge and energy diagnostics. This procedure decouples the task of micrometer-precision spatial alignment of the injector laser from femtosecond-level temporal synchronization additionally required for the plasma photocathode. When the laser energy is reduced to a certain level, plasma torch injection also ceases to function because the plasma blowout is not sufficiently deformed by the induced density perturbation. In contrast, helium electrons are trapped even at reduced laser energies when the laser pulse releases them directly inside the plasma blowout, provided a sufficiently high trapping potential exists (see Methods). These transitions are the key to access the plasma photocathode regime and allow to deploy the two injection modes sequentially: starting from plasma torch injection as stepping stone, a timing delay scan, combined with reduced injection laser energy, isolates and reveals the plasma photocathode process as summarized in Figure 3. Figure 3 | Injected charge as a function of laser energy and timing. a, 5 mJ laser energy (plasma torch injection regime). b, 1 mJ laser energy (mixed mode regime). c, 0.5 mJ laser energy (plasma photocathode injection regime). The blue crosses represent charge values measured on the downstream spectrometer, and the light blue/dark blue bars indicate the maximum/average charge per time bin, respectively. The corresponding values obtained from PIC-simulations (see Methods) are shown in red. The red open circles in a show that the range of injected charge output can be reproduced by variations of channel width and/or alignment. For the grey areas in b and c, no data was collected. Experimental data are sorted via EOS time stamping to account for the system-inherent time-of-arrival (TOA) jitter (see Methods). As shown in Fig. 3a, the observed injected charge exhibits a plateau when a 5 mJ laser pulse arrives prior to the driver beam, characteristic of the plasma torch injection mode. When reducing the injection laser energy to 1 mJ, the plasma filament is weakened and down-ramp injection loses its effectiveness. This leads to an intermediate mixed mode regime with reduced injected charge for early laser pulse arrival times and a pronounced peak when the laser pulse arrives immediately after the driver electron", + "the injection laser energy to 1 mJ, the plasma filament is weakened and down-ramp injection loses its effectiveness. This leads to an intermediate mixed mode regime with reduced injected charge for early laser pulse arrival times and a pronounced peak when the laser pulse arrives immediately after the driver electron beam (Fig. 3b). Finally, a further reduction of laser energy to 0.5 mJ reveals injected charge only in a narrow time window, consistent with the duration of the overlap of the laser with the plasma wave (Fig. 3c). This is the sought-after plasma photocathode injection mode. An imaging spectrometer downstream of the plasma source allows measurement of electron beam charge, spot size and energy spectrum. Figure 4 presents spectrometer data for representative plasma photocathode witness bunches. Figure 4a shows energy-sorted spectra from ~0.3 GeV to ~0.7 GeV. This range agrees with simulations that investigate details of the acceleration process within the plasma channel (see Methods and Extended Data Figure 2). Close to the spectrometer imaging energy of 0.5 GeV, measurement in the dispersed plane yields a minimum energy spread of 2.1 ± 0.3 % (r.m.s.), as shown in Fig. 4b. Far from this imaging energy, measurement in the non-dispersed plane yields a divergence of 380 ± 30 µrad (r.m.s.), as shown in Fig. 4c. Combining the measured divergence with a calculated beta- function of 1.5 cm at the exit of the plasma results in a minimum normalized emittance of n  1 mm mrad (see Methods and Extended Data Fig. 3), consistent with simulations. Figure 4 | Spectrometer data of representative electron bunches measured in the plasma photocathode regime from Fig. 3c. a, the energy-sorted spectra for shots with charge higher than 5 pC and TOA > 0 with colour-coded charge density. b, a selected shot with energy close to imaging energy of 0.5 GeV, showing a minimum energy spread of 2.1 ± 0.3 % (r.m.s.). c, a selected shot with energy far from the imaging energy, indicating a horizontal divergence of 380 ± 30 µrad (r.m.s.), and estimated emittance of n  1.5 mm mrad. The stability and quality of electron beam production is limited by the plasma channel width, and incoming laser and electron driver beam jitter encountered in the experiment. Simulations show that the plasma channel is a technical bottleneck responsible for injected charge jitter and limited energy gain (see Methods). A wider plasma channel, better pointing stability and an order of magnitude better timing accuracy of driver beam and injection laser are technically feasible and will permit to improve injected charge stability and energy gain substantially. At the same time, a wider plasma channel allows operation at lower plasma densities and in collinear geometry, which also improves beam quality in terms of residual energy spread, emittance and brightness12. In summary, we demonstrate controlled electron bunch generation in an electron-driven plasma wakefield accelerator by a decoupled injection laser pulse, releasing electrons at ionization threshold from an overlaid gas component. This allows to realize two complementary injection modes and to seamlessly switch between them: the plasma torch down-ramp injection mode", + "In summary, we demonstrate controlled electron bunch generation in an electron-driven plasma wakefield accelerator by a decoupled injection laser pulse, releasing electrons at ionization threshold from an overlaid gas component. This allows to realize two complementary injection modes and to seamlessly switch between them: the plasma torch down-ramp injection mode is used to locate and then to access the plasma photocathode regime. The experimental introduction of the plasma photocathode paves the way for production of electron beams with nm-rad-level normalized emittance and brightness up to four orders of magnitude higher than state-of-the-art. Such beams may allow realization of compact, hard x-ray light sources such as free-electron lasers with unprecedented gain12 characteristics, as well as testing and optimization of emittance preservation for potential future plasma-based linear collider stages. METHODS Electron driver beam. The SLAC linac delivers electron beams with a charge up to Q  3.2 nC and energy of W  20.35 GeV  2 % FWHM over a length of approximately 2 km at a repetition rate up to 10 Hz to the FACET experimental area (see Extended Data Fig. 1). The beams are longitudinally compressed in a magnetic chicane to lengths of z  20-40 µm (r.m.s.), and are transversally focused by the five quadrupole magnets of the final focus system to x  25 µm (r.m.s.) and y  30 µm (r.m.s.), with corresponding beta function of x  25 cm and y  100 cm. Plasma source. The entire ~5.4 m long section between the upstream 50 µm-thick Beryllium foil and a 100 µm-thick diamond foil downstream is filled with a hydrogen/helium gas mixture at densities of nH2  nHe  0.65  1017 cm-3. As shown in Extended Data Fig. 1, a Ti:sapphire laser pulse with energy of 170 mJ and pulse duration of 55 fs (FWHM) is focused by an axilens with focal length of 3 m and depth of focus 1 m long, generating a longitudinal Bessel beam intensity profile along the electron driver beam axis. Optical Transition Radiation (OTR) screens are used to align the laser with the electron beam axis. This laser pulse producing an intensity distribution with peak intensity of 3  1014 W cm-2, sufficient to exceed the tunneling ionization threshold of hydrogen, arrives ~20 ps before the electron driver beam. The axially symmetric hydrogen plasma profile produced, as calculated with the Ammosov-Delone- Krainov (ADK) model22, is shown in Extended Data Fig. 2a. The plasma initiates at the longitudinal position of z  0.05 m in Extended Data Fig. 2a, and extends over a length of z  70 cm. Transversally, it is tapered in a complex manner as result of the Bessel profile, extending to the full width r  130 µm at the widest position. The exponential character of the tunneling ionization rate leads to a sharp transition from gas to fully ionized plasma with peak hydrogen plasma densities of ne,H  1.3  1017 cm-3. This choice of density is the result of various considerations. Generally, one would want to work at much lower densities, because then the", + "the tunneling ionization rate leads to a sharp transition from gas to fully ionized plasma with peak hydrogen plasma densities of ne,H  1.3  1017 cm-3. This choice of density is the result of various considerations. Generally, one would want to work at much lower densities, because then the plasma blowout is much larger, and it is much easier to stably hit the desired position within the blowout with the injection laser for a technically given absolute laser-electron-beam jitter in space and time. This strongly enhances reproducibility of the output beam. At the same time, a transverse kick by the drive beam (see Fig. 1d-f), which increases the transverse emittance of the witness bunch, would be avoided, and a reduced residual energy spread would be obtained12. However, the plasma blowout must fit within the preionized plasma channel of limited and varying width. This limit is not fundamental, but was experimentally imposed by the available laser system energy and available space. Even with an experimentally optimized density of ne,H  1.3  1017 cm-3, the blowout touches the boundaries of the channel for most of the acceleration, which compromises the blowout strength and strongly limits the energy gain (Extended Data Fig. 2a). An upper hydrogen plasma density limit for the given FACET electron beam results from unwanted helium ionization and dark current by the more strongly pinching driver beam fields and wakefields7,8,21, which sets in approximately at ne,H > 2  1017 cm-3. It should be noted that in contrast to purely laser-based ionization injection schemes23,28, the injection in electron beam-driven approaches arises from the independently tunable helium density in the mixture. Adjusting the helium density is therefore an independent knob which can be used to tune the injected charge from femto-Coulomb to nano-Coulomb levels. Laser- based injection schemes such as23,28-30 are showing transformative impact on optimizing laser- plasma wakefield acceleration, which in turn could lead to production of electron beams employable as drivers for beam-driven plasma wakefield acceleration9 in ultracompact hybrid systems. Probe and plasma photocathode injection lasers. A laser pulse with duration of 65 fs (FWHM) is split up into two pulses that traverse optical paths which are independently tunable in energy and equipped with delay stages for temporal synchronization. One collimated laser arm serves an upstream electro-optic sampling (EOS) diagnostic for shot-to-shot time-of-arrival (TOA) measurement of driver electron beam-to-laser relative timing, while the plasma photocathode injection laser arm is focused by an f/22.9 off-axis-parabolic mirror to a spot size of ~ 20 µm perpendicular to the electron beam axis. The upstream OTR screen is used for spatial alignment of the focused injection laser and the driver beam (see Extended Data Fig. 1). The EOS system non-destructively measures shot-to-shot-jitter between the laser and electron driver beam as 109 ± 12 fs (r.m.s.), and simultaneously records the TOA for the injection experiments with a resolution of 25 fs. A CCD camera beneath the beam line images the laser-generated plasma filament across the electron driver beam axis. These diagnostics combined facilitate the alignment and synchronization of the injection laser and electron-driven plasma", + "fs (r.m.s.), and simultaneously records the TOA for the injection experiments with a resolution of 25 fs. A CCD camera beneath the beam line images the laser-generated plasma filament across the electron driver beam axis. These diagnostics combined facilitate the alignment and synchronization of the injection laser and electron-driven plasma wake. Electron witness beam generation and measurement. In an unperturbed plasma wave, the trapping condition7,23 is given by (max-)/[(m0c2/e)(1-γph-1)] <-1, where  is the electrostatic trapping potential, m0 and e are the electron rest mass and charge, respectively, c is the speed of light and γph is the relativistic plasma wave Lorentz factor. This condition is easily fulfilled by the strong plasma wave at FACET (see Supplementary Movies). As shown in Extended Data Fig. 2, electrons injected at around zinj  0.2 m are rapidly accelerated by tens of GV m-1 fields in this comparably wide part of the plasma. Due to the transversely tapered plasma profile, however, the wakefields are progressively reduced in z and the wave geometry elongates. Witness electrons are then partially located in the decelerating phase of the plasma wave, which limits the total energy gain. Extended Data Fig. 2b shows the expected range of final witness electron energies of up to ~1.9 GeV for the given plasma profile with consideration of possible injection positions in the lab frame and the trapping positions in the co-moving frame. It is in good agreement with experimental data, which have been obtained by the imaging spectrometer after transporting the electrons downstream in a well-characterized beam line. The integrated probability of injecting electrons, a measure of expected witness charge, is highest for injection that results in production 0.3-0.7 GeV electrons. A wider pre-ionized plasma channel would be suitable to resolve issues of injected electron jitter and energy gain limitations. After exiting the plasma and entering the vacuum section through the diamond window (see Extended Data Fig. 1), the electrons are captured and focused by a quadrupole doublet, and dispersed by a dipole magnet onto a CCD-monitored phosphor screen located 22 m after the end of the plasma. Charge is measured by BPMs (with an accuracy of 4-5%) and also by the signal intensity on the phosphor screen (calibrated based on BPM measurements, in combination having an accuracy of ~10%). Extended Data Fig. 3 summarizes the analysis of energy spread and divergence measurements. The energy spread measured on the spectrometer has an additional contribution from the vertical emittance, but close to the imaging energy of 0.5 GeV the emittance contribution is negligible, thus resolving the actual energy spread (see Extended Data Fig. 3a). To derive the divergence, Gaussian fits of the spectral charge density in the transverse (non-dispersive) direction are combined with beam transport simulations including multiple Coulomb scattering24 in the exit foil. The corresponding emittance is calculated with a beta function of 1.5 cm, obtained by integrating Hill’s equation through the modeled plasma density profile. The measured transverse beam sizes and estimated emittances are shown in Extended Data Fig. 3b. Plasma wakefield acceleration simulations. The fully explicit 3D particle-in-cell (PIC) code VSim25", + "The corresponding emittance is calculated with a beta function of 1.5 cm, obtained by integrating Hill’s equation through the modeled plasma density profile. The measured transverse beam sizes and estimated emittances are shown in Extended Data Fig. 3b. Plasma wakefield acceleration simulations. The fully explicit 3D particle-in-cell (PIC) code VSim25 is used to simulate the driver beam-plasma interaction and the injection processes depicted in Fig. 1, Fig. 3 and the Supplementary Movies. An accurate representation of the pre- ionized plasma profile (see Extended Data Figure 2a) is directly loaded into the simulations. The electron driver beam has been implemented as a tri-Gaussian beam based on the experimental measurements of x, y, z, charge, energy, energy spread and emittance. The beam was initialized at the entrance of the plasma. The simulation box has a size of 500 µm  300 µm  300 µm in the longitudinal dimension and the two transverse dimensions, respectively. An additional 16 µm on each transverse border are reserved as absorbing boundaries via perfectly matched layers. The number of the cells for the simulation box is 250  166  166. The coordinate system represents a co-moving frame at the speed of light. Eight macro-particles per cell (PPC) are used to model the plasma, and 16 PPC for the driver beam, respectively. Time-resolved PIC simulations are needed to resolve the injection processes. Hence, the number of particles and the spatial resolution of the simulations are chosen as a trade-off between desired accuracy and the long propagation distance. All simulations use the Yee scheme for electro-magnetic field updates, where the time-step is chosen to be 0.5 times the Courant limit. Charge and current deposition is applied according to the Esirkepov algorithm in 3rd order, and the Boris method is used to push the macroparticles. The driver beam and wakefield evolution are simulated throughout the plasma, allowing the mapping of the accelerating field and trapping conditions over the whole propagation distance to estimate the expected energy gain (see Extended Data Figure 2). The configuration at the experimental injection point zinj  0.2 m inside the plasma (see Extended Data Fig. 2a) serves as the initial state of the individual injection simulations shown in Fig. 1 and the Supplementary Movies, as well as for the data points in Fig. 3. Simulations are performed with different laser energies and relative laser-to-beam timing, and the trapped charge in the wake is presented in Fig. 3, where zero is defined as the time when the electron and laser beam centres cross each other. Three complementary simulation approaches are used to cover the large delay range. For positive timing values, the injection laser is added as an envelope in paraxial approximation, and tunneling ionization is calculated based on an averaged ADK model26,27. For negative timing values, the electron driver beam is externally loaded into a longer simulation box, allowing the laser to dynamically generate the plasma filament before the driver beam arrival. For even larger delays (< -1 ps), the fully formed plasma filament is loaded in the simulation. The injection simulations were performed 5", + "values, the electron driver beam is externally loaded into a longer simulation box, allowing the laser to dynamically generate the plasma filament before the driver beam arrival. For even larger delays (< -1 ps), the fully formed plasma filament is loaded in the simulation. The injection simulations were performed 5 mm beyond the injection point and the same spatial and energy cuts were applied for all cases to discard the low energy and high divergence electrons which would not survive the downstream acceleration and beam transport. All simulation approaches are consistent and in good agreement with the experimentally obtained charge values, as shown in Fig. 3. We note that for the highest energy considered (5 mJ), a perfectly Gaussian laser pulse may even ionize the second level of helium. Simulations including this additional ionization level confirm that the general injection behavior is maintained albeit at increased charge values. This experimental possibility of second level ionization can account for occasionally higher charge values as shown in Fig. 3(a). Supplementary Movies visualize the two injection modes shown in Fig. 1. Extended Data Figure 1 | Overview of experimental setup at SLAC FACET. The 2 km linac-generated electron driver beam arrives from the top left direction, and a plasma channel is generated by focusing a high-power laser pulse along the electron beam axis with an axilens. A separate laser pulse feeds an electro-optical sampling (EOS) unit upstream of the interaction point, and the plasma photocathode laser is focused onto the electron beam axis at an angle of 90° by an off-axis parabolic mirror. Beryllium (Be) and diamond windows confine the gas to the plasma source region. Optical transition radiation (OTR) screens are used to align the lasers with the electron beam axis. Beam position monitors (BPMs) upstream and downstream of the plasma source are used for charge measurement, and a beam transport line with an imaging spectrometer is used to measure and derive the electron bunch energy, energy spread and divergence. Extended Data Figure 2 | Calculated plasma source profile, longitudinal electric field and energy range of accelerated electrons. a, the hydrogen plasma density profile produced by the axilens-focused laser pulse along the electron driver beam axis z. The dashed red line at z  0.2 m indicates the position of the transverse injection laser pulse in the laboratory frame. The varying width of the plasma source distorts the plasma oscillation trajectories and thus modulates the blowout strength and shape along z. This feature leads to a varying longitudinal electric field Ez during the propagation through the plasma; the evolution of the longitudinal field depends on the specific positions =z-ct of the witness electrons in the co-moving frame. The black line shows the simulated Ez (right y-axis) sampled by the electrons trapped at trap  -107 µm during acceleration, where  = 0 is defined by the centre of the electron driver beam in the co-moving frame as in Figure 1. b, the expected final energy range of the accelerated electrons (colour-coded) arising from different injection positions in the laboratory frame around zinj  0.2", + "-107 µm during acceleration, where  = 0 is defined by the centre of the electron driver beam in the co-moving frame as in Figure 1. b, the expected final energy range of the accelerated electrons (colour-coded) arising from different injection positions in the laboratory frame around zinj  0.2 m and corresponding possible trapping positions trap from -107 µm to - 139 µm. In the underlying simulations of the propagating blowout evolution, electrons that move into the defocusing phase of the blowout at any time do not survive the acceleration process, resulting in the white gap. A wider plasma channel allows to harness constantly accelerating electric fields, which then implies tens of GeV energy gain over the length of the plasma channel6. Extended Data Figure 3 | Statistical analysis of beam quality for the shots in Figure 4 (a). a, the experimentally measured apparent r.m.s. energy spreads (blue data points), as a result of the actual energy spread (grey lines) and an additional vertical emittance contribution (green lines). A minimum energy spread of 2 % is obtained for the shot shown in Fig. 4b. b, the corresponding r.m.s. transverse beam sizes in the non-dispersed plane with emittance estimation (green contours) using a beta function of 1.5 cm. A shot with emittance of 1.5 mm mrad is shown in Fig. 4c. The grey area shows a divergence limit of 1 mrad imposed by the aperture of a laser out-coupling mirror having a central hole for beam passage (located between the second OTR and the diamond window, not shown in Extended Fig. 1). The black lines represent the spectromter limits according to a simulated zero emittance and zero energy spread of the beam. Shots with compromised data characteristics are omitted. Data availability Data associated with research published in this paper will be available at publication. References 1 Tajima, T. & Dawson, J. M. Laser electron accelerator. Phys. Rev. Lett. 43, 267-270 (1979). 2 Leemans, W. P. et al. Multi-GeV Electron Beams from Capillary-Discharge-Guided Subpetawatt Laser Pulses in the Self-Trapping Regime. Phys. Rev. Lett. 113, 245002 (2014). 3 Chen, P., Dawson, J. M., Huff, R. W. & Katsouleas, T. Acceleration of electrons by the interaction of a bunched electron beam with a plasma. Phys. Rev. Lett. 54, 693-696, (1985). 4 Blumenfeld, I. et al. Energy doubling of 42 GeV electrons in a metre-scale plasma wakefield accelerator. Nature 445, 741-744, (2007). 5 Rosenzweig, J. B. et al. Experimental-Observation of Plasma Wake-Field Acceleration. Phys. Rev. Lett. 61, 98-101 (1988). 6 Litos, M. et al. High-efficiency acceleration of an electron beam in a plasma wakefield accelerator. Nature 515, 92-95 (2014). 7 Oz, E. et al. Ionization-induced electron trapping in ultrarelativistic plasma wakes. Phys. Rev. Lett. 98, 084801 (2007). 8 Vafaei-Najafabadi, N. et al. Beam Loading by Distributed Injection of Electrons in a Plasma Wakefield Accelerator. Phys. Rev. Lett. 112, 025001 (2014). 9 Hidding, B. et al. Ultracold electron bunch generation via plasma photocathode emission and acceleration in a beam-driven plasma blowout. Phys. Rev. Lett. 108, 035001(2012). 10 Li, F. et al. Generating high-brightness electron beams via ionization", + "Injection of Electrons in a Plasma Wakefield Accelerator. Phys. Rev. Lett. 112, 025001 (2014). 9 Hidding, B. et al. Ultracold electron bunch generation via plasma photocathode emission and acceleration in a beam-driven plasma blowout. Phys. Rev. Lett. 108, 035001(2012). 10 Li, F. et al. Generating high-brightness electron beams via ionization injection by transverse colliding lasers in a plasma-wakefield accelerator. Phys. Rev. Lett. 111, 015003(2013). 11 Wittig, G. et al. Optical plasma torch electron bunch generation in plasma wakefield accelerators. Phys. Rev. ST Accel. Beams 18, 081304 (2015). 12 Manahan, G. G. et al. Single-stage plasma-based correlated energy spread compensation for ultrahigh 6D brightness electron beams. Nat. Commun. 8, 15705 (2017). 13 Bostedt, C. et al. Linac Coherent Light Source: The first five years. Rev. Mod. Phys. 88, 015007 (2016). 14 Rosenzweig, J. B. & Colby E. Charge and wavelength scaling of RF photoinjector designs. AIP Conference Proceedings 335, 724 (1995). 15 Hogan, M. J. et al. Plasma wakefield acceleration experiments at FACET. New J. Phys. 12, 055030 (2010). 16 Davidson N., Friesem A. A., and Hasman E. Holographic axilens: high resolution and long focal depth. Opt. Lett. 16, 523-525 (1991). 17 Green, S. Z. et al. Laser ionized preformed plasma at FACET. Plasma Phys.Control. Fusion 56, 084011 (2014). 18 Bulanov, S., Naumova, N., Pegoraro, F., and Sakai, J. Particle injection into the wave acceleration phase due to nonlinear wake wave breaking. Phys. Rev. E 58, 5257-5260 (1998). 19 Suk, H., Barov, N., Rosenzweig, J. B. & Esarey, E. Plasma electron trapping and acceleration in a plasma wake field using a density transition. Phys. Rev. Lett. 86, 1011- 1014(2001). 20 Geddes, C. G. et al. Plasma-density-gradient injection of low absolute-momentum- spread electron bunches. Phys. Rev. Lett. 100, 215004 (2008). 21 Manahan, G. G. et al. Hot spots and dark current in advanced plasma wakefield accelerators. Phys. Rev. Accel. Beams 19, 011303 (2016). 22 Ammosov, M. V., Delone, N. B. & Krainov, V. P. Tunnel ionization of complex atoms and atomic ions in a varying electromagnetic-field. Sov. Phys. JETP 64, 6 (1986). 23 Pak, A. et al. Injection and trapping of tunnel-ionized electrons into laser-produced wakes. Phys. Rev. Lett. 104, 025003 (2010). 24 Chao, A. W. Handbook of Accelerator Physics and Engineering. Second edition. World Scientific, Singapore, 2013. 25 Nieter, C. & Cary, J. R. VORPAL: a versatile plasma simulation code. J. Comput. Phys. 196, 448-473 (2004). 26 Bruhwiler, D. L. et al. Particle-in-cell simulations of tunneling ionization effects in plasma-based accelerators. Phys. Plasmas 10, 2022-2030 (2003). 27 Chen, M. et al. Numerical modeling of laser tunneling ionization in explicit particle-in- cell codes. J. Comput. Phys. 236, 220-228 (2013). 28 Umstadter, D. et al. Laser Injection of Ultrashort Electron Pulses into Wakefield Plasma Waves. Phys. Rev. Lett. 76, 2073 (1996). 29 Faure, J. et al. Controlled injection and acceleration of electrons in plasma. Nature 444, 737–739 (2006). 30 Thomas, A.G.R. et al. Monoenergetic Electronic Beam Production Using Dual Collinear Laser Pulses. Phys. Rev. Lett. 100, 255002 (2008). Supplementary Information is available in the online version of the paper. Acknowledgements The FACET E210 plasma wakefield", + "al. Controlled injection and acceleration of electrons in plasma. Nature 444, 737–739 (2006). 30 Thomas, A.G.R. et al. Monoenergetic Electronic Beam Production Using Dual Collinear Laser Pulses. Phys. Rev. Lett. 100, 255002 (2008). Supplementary Information is available in the online version of the paper. Acknowledgements The FACET E210 plasma wakefield acceleration experiment was built and operated with support from UCLA (US DOE contract DE-SC0009914), RadiaBeam Technologies (DOE contract DE-SC0009533), the FACET E200 team and the U.S. Department of Energy under contract number DE-AC02-76SF00515, H2020 EuPRAXIA (Grant No. 653782), Helmholtz VI, EPSRC (Grant No. EP/N028694/1), and the Research Council of Norway (Grant No. 230450). This work used computational resources of the National Energy Research Scientific Computing Center, which is supported by DOE DE-AC02-05CH11231, of JURECA (Project hhh36), of HLRN and of Shaheen (Project k1191). D.L.B. acknowledges support of the US DOE Office of High Energy Physics under Award No. DE-SC0013855. Author Contributions All authors contributed extensively to the work presented in this paper. Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of the paper. Correspondence should be addressed to bernhard.hidding@strath.ac.uk", + "Electron Beam Production by Pyroelectric Crystals James D. Brownridgea Department of Physics, Applied Physics, and Astronomy State University of New York at Binghamton, P.O. Box 6016 Binghamton, New York 13902-6016 Stephen M. Shafrothb Physics and Astronomy Department, University of North Carolina at Chapel Hill, Chapel Hill North Carolina 27599-3255 Abstract Pyroelectric crystals are used to produce self-focused electron beams with energies greater than 170 keV. No high voltage power supply or electron gun is needed. The system works by simply changing the temperature of a crystal of LiNbO3 or LiTaO3 by about 100oC in dilute gas. Electron beam energy spectra as well as positive-ion-beam energy spectra and profiles are shown. A change in the crystal temperature of 100oC will cause a spontaneous change in polarization. The change in polarization will be manifested by a change in charge on the surface of the crystal. It is this uncompensated charge that produces the electric field, which accelerates the electrons, or the positive ions and gives rise to the plasma, which in turn focuses them. The source of the accelerated electrons or positive ions is gas molecules ionized near the crystal surface. When the crystal surface is negative electrons are accelerated away from it and positive ions are attracted to the surface. These positive ions reduce the net negative charge on the surface thereby reducing the electric field, which causes the electron energy to decrease over time even though the focal properties remain unchanged. When the surface is positive the reverse obtains and the positive ion beam energy decreases over time as well. We will present video clips, photographic and electronic data that demonstrate many of the characteristics and applications of these electron beams. Introduction Even though pyroelectric crystals have been known since ancient Greek times1, surprising new effects and practical applications are constantly being discovered 1-9. There is a voluminous literature on electron emission by pyroelectric crystals. Reference 10, a comprehensive review article and (11-15) are representative but none of them suggests that energetic electron beams are produced by such crystals. Much of the current industrial use of these crystals is in infrared detection, sensitive temperature change detectors and photonics, e.g., the detection of THz radiation16 in LiNbO3. For these reasons few studies of the behavior of these crystals in dilute gases have previously been done and so no one has previously discovered the phenomena being reported on here. The first known naturally occurring pyroelectric crystals were kidney stones of lynx1. All pyroelectric crystals including artificially produced LiNbO3 and LaTaO3 are asymmetric crystals. They become electrically charged on heating and cooling and give rise to electric fields at their surfaces of up to 106 V/cm18. The opposite side of a crystal which is heated from its + z base becomes positively charged on heating and negatively charged on cooling. This is due to the movement of Li+ and Nb- ions relative to the oxygen lattice12. If the crystal is immersed in a dilute gas the net charge on the crystal surface at any time will depend on the gas type and pressure as well", + "and negatively charged on cooling. This is due to the movement of Li+ and Nb- ions relative to the oxygen lattice12. If the crystal is immersed in a dilute gas the net charge on the crystal surface at any time will depend on the gas type and pressure as well because when the surface is negative, positive ions from the gas will bombard it and when the surface is positive, electrons from the ionized gas will also bombard it. Crystal x-ray emission occurs on heating and target x-ray emission on cooling when the -z base is exposed and light is produced both on heating and on cooling in the plasma adjacent to the crystal. Chronologically, JDB2 reported on the production of characteristic x-rays by cesium nitrate crystals in 1992. Then in 1999 he and S. Raboy wrote a comprehensive paper3 describing the x- ray emission and light production in the plasma for several different crystals and gases. Also in 1999 Shafroth et. al. studied the time dependence of x-ray emission4. Further they studied the effect on total x-ray yield for five different gases and found a marked decrease in yield for O2 which was due to the production of ozone in the plasma. Also in 1999 Shafroth and Brownridge reported on the use of pyroelectric crystals in the teaching of x-ray physics5. Then in 2001 JDB and collaborators reported on the production of energetic electrons by pyroelectric crystals in dilute gases6. They found that multiple, energetic (~100 keV) nearly monoenergetic electrons were produced on cooling these crystals when the - z base faced the detector. Later JDB and others7 found that the maximum energy of the emitted electrons depended somewhat on gas type but strongly on pressure. Then JDB and SMS found8 that if the crystals were ground to a cylindrical shape that focused beams of electrons were produced as the crystal cooled and focused beams of positive gas ions were produced as the crystal was heated. Most recently, S. Bedair et. al., using a LabVIEW program have studied the effect of pressure on the time dependent yield of crystal x-rays and target current9. Fig. 1. (a) Photograph of the experimental arrangement with ZnS screen, crystal, and heater resistor, (b) Beam spot at ~0.5 mtorr, (c) ~3 mtorr, (d) ~8 mtorr, where the beam blows up. Note how the beam is defocused as pressure increases. Focused Electron Beams This report describes the production of self-focusing electron beams arising from near pyroelectric crystal surfaces in dilute gases after the crystal has been heated and returned to room temperature. In one case a 5 mm diam x 5 mm cylindrical LiNbO3 crystal in an atmosphere of <10 mtorr of dry N2 was contained in a glass tube. The crystal was heated to about 115 0C and then allowed to cool to room temperature. Heat was supplied by passing ~100 mA through a wire-wound 60 ohm resistor which was epoxied to the crystal's + z base. After cooling to room temperature, spatially-stable electron beams were produced. They were made visible by placing a ZnS screen", + "0C and then allowed to cool to room temperature. Heat was supplied by passing ~100 mA through a wire-wound 60 ohm resistor which was epoxied to the crystal's + z base. After cooling to room temperature, spatially-stable electron beams were produced. They were made visible by placing a ZnS screen at the focal length (17 mm) of the crystal as illustrated in Fig. 1(a), which shows a photograph of the ZnS screen; the cylindrical crystal and the 60 ohm wire-wound resistor used to heat the crystal. We studied the effect of pressure on the beam. Typical results at ~0.5 mtorr 1(b), ~3 mtorr 1(c) and ~8 mtorr 1(d) are shown in Fig 1(b), 1(c), 1(d) respectively. As the pressure increased, the beam spot became more diffuse and brighter. At ~8 mtorr the beam blew up and its intensity dropped to zero. The dynamical behavior of the electron beam with pressure, which is illustrated in the video clip17 suggests a gas-multiplication effect. The focusing effect may be due to plasma focusing. All results reported on in this paper were repeated many times. Fig. 2. Electron Energy Spectra taken with a 10 mm x 4mm dia LiNbO3 crystal in 2.7 mTorr of N2 . Fig. 3. Electron energy spectrum for highest energy electron beam yet obtained. Same conditions as Fig. 2. Fig. 4. Schematic diagram of the experimental arrangement for scanning the electron beam. The slit and detector move. The dotted lines show an idealized cartoon of the crystal focused electron beam. Next a 100 μm surface-barrier electron detector was installed in place of the ZnS screen so that electron energy spectra could be recorded. The time- dependent electron-beam energy spectrum, which was taken in \"snapshot mode\", i.e. over an interval of <60 s, typically, exhibited discrete peaks at integral multiples of the lowest energy peak. This indicated that nearly monoenergetic multiple electron production6 was occurring. Typical electron energy spectra are shown in Fig. 2. where the multiple peak effect due to pile up is shown at three different crystal temperatures. Fig. 3. shows the highest energy electron spectra so far obtained. The experimental arrangement for scanning the electron beam profile is 0 50 100 150 200 250 300 350 400 450 500 550 600 0 2000 4000 6000 ~372 keV ~248 keV ~124 keV Temp. 25 oC keV Counts 0 50 100 150 200 250 300 350 400 450 500 550 600 0 4000 8000 12000 16000 ~298 keV ~447 keV ~149 keV Temp. 44 oC Counts 0 50 100 150 200 250 300 350 400 450 500 550 600 0 2000 4000 6000 8000 ~381 keV ~254 keV ~127 keV Temp. 65 oC Counts 0 100 200 300 400 500 600 700 800 900 1000 0 5000 10000 15000 20000 25000 30000 ~680 keV ~511 keV ~340 keV ~170 keV Counts Electron energy (keV) shown schematically in Fig. 4. The crystal and heater assembly could be moved toward or away from the detector, which was useful for checking the focusing effect. Fig. 5. Beam scans taken at different crystal-to- slit distances. Each", + "~680 keV ~511 keV ~340 keV ~170 keV Counts Electron energy (keV) shown schematically in Fig. 4. The crystal and heater assembly could be moved toward or away from the detector, which was useful for checking the focusing effect. Fig. 5. Beam scans taken at different crystal-to- slit distances. Each scan lasted ~30 s. The focal length is ~16.5 mm. Profiles were taken in the order shown. Fig. 6. Electron beam profiles taken 16 hours later. The focal length is still ~16.5 mm. The beam intensity is down after 16 hours of continuous electron emission, however it is still focused. The detector was covered by a lead screen with a 0.1mm slit so that the beam profile could be obtained by moving the slit-detector assembly at constant speed across the beam at different crystal-to-slit distances. Fig. 5. shows results of ~ 30 s scans taken at different crystal-to-slit distances. Fig. 7. The electron beam count rate profile for a 10 mm x 4 mm LiNbO3 crystal (open circles). The electron beam energy (solid circles) is shown for each slit position. Note the linear decrease in energy as the scans progress and as time increases. (See Fig 8 for comparison). All pulses corresponding to electron energies > 15 keV were counted for these scans. From this figure it can be seen that the best focus is obtained 16.5 mm crystal-to-slit distance. Fig. 6. shows electron beam scans taken 16 hours later. The focal length is still 16.5 mm. Fig. 7. obtained with a 4mm diam x 10 mm LiNbO3 crystal shows the electron beam count rate profile as the detector and slit are scanned across the beam at a distance of 20 mm. The FWHM is about 4 mm. Also shown is the electron energy at each slit position. The uniform decrease in electron energy with slit position and time is similar to that described below. Electron Beam Energy Spectra Once the crystal reached room temperature the beam energy decreased nearly exponentially with time (Fig. 8); faster with increasing pressure. In order to produce higher energy electrons the 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 Counts Counts Counts Counts 22mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 21mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 20mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 19mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 18mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0", + "0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 18mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 17mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 16mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 15mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 14mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 13mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 12mm 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 0 25 50 75 10 0 Slit/detector position as it is moved across the beam Focal length 16.5mm 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 12 mm 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 14 mm 13 mm 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 15 mm 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 16 mm 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 17 mm 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 Counts Counts Counts Slit/detector position as it is moved across the beam 18 mm 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 19 mm 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 20 mm 0 1 2 3 4 5 6 7 8 9 10 11 30 35 40 45 50 55 60 65 70 Slit position (mm) Electron beam energy (keV) 20 40 60 80 100 120 140 160 Electron count rate (c/sec) crystal was heated from the + z base to about 160 0C and allowed to cool to room temperature at <10 mtorr. Fig. 8. Electron beam energy as a", + "65 70 Slit position (mm) Electron beam energy (keV) 20 40 60 80 100 120 140 160 Electron count rate (c/sec) crystal was heated from the + z base to about 160 0C and allowed to cool to room temperature at <10 mtorr. Fig. 8. Electron beam energy as a function of time after the crystal has reached room temperature. Note the nearly exponential decay of the beam energy. The maximum energy electrons (170 keV) were produced after the crystal temperature had dropped to ~30 0C. Independent proof that the electron beam energy was at least 80 keV (Au K edge) for a significant time was confirmed by irradiating gold and observing the K X- rays2,3. Gold K electrons were being ionized thereby producing Au K X-rays as long as the incident electron energy was greater than 80 keV i.e. starting as the crystal cools from 160 0C when the increasing electron energy reached 80 keV and continuing until the maximum energy (170 keV) is reached and then until the electron energy falls to less than 80 keV, sometime after the crystal has reached room temperature. Phenomenological Description of These Processes If the crystal is subjected to a rapid temperature change the polarization will be manifested briefly by the presence of an electric field whose strength is proportional to the surface charge density, which in turn is a function of the change in temperature and depends on the ambient pressure. When the crystal is in a reduced pressure environment and is subjected to the same temperature change, the polarization change is manifested by an electric field that lasts much longer; at a pressure of about 10-6 torr, it takes more than 30 hours for the resulting surface charge to be neutralized, whereas at 1 atm neutralization is virtually instantaneous3. The neutralization of the electric field produced by a change in polarization is due to bombardment of and attachment to the surface by positive ions. To summarize, as long as the net surface charge is sufficiently negative, electrons are accelerated away from the crystal in a focused beam whose energy can be up to 170 keV after which the energy of the electrons in the beam decreases nearly exponentially with time until the electrons become undetectable. It is remarkable that the beam becomes spatially stable once the crystal has reached room temperature, even before in some cases. Positive Ion Beams We believe that singly charged positive ion beams arising from the residual gas have also been observed. For practical reasons they can best be studied by exposing the + z base of the crystal and taking measurements as the crystal cools, although the same phenomena can be observed on heating a crystal with its -z base exposed. Figure 9 shows an energy spectrum obtained over a period of 35 s with a crystal of LiNbO3, 10 mm x 4mm dia heated to 1700C after the crystal had 0 25 50 75 100 125 150 175 200 225 250 275 300 325 1 10 100 Electron energy (keV) Time (minutes) Fig. 9. Energy spectrum", + "obtained over a period of 35 s with a crystal of LiNbO3, 10 mm x 4mm dia heated to 1700C after the crystal had 0 25 50 75 100 125 150 175 200 225 250 275 300 325 1 10 100 Electron energy (keV) Time (minutes) Fig. 9. Energy spectrum (30 s snapshot) of N2+ ion beam obtained with a. LiNbO3 crystal on cooling from 170 0C. The + z base is exposed. The spectrum was obtained after cooling to 35 0C. The gas was N2. Fig. 10. The N2+ ion beam count rate profile (open circles) and the beam energy (solid circles) at each slit position. This should be compared with the electron beam results in Fig. 7, where the beam energy is unaffected by the slit being at the focal position. cooled to ~350C . Here the gas was N2 so the beam was presumably N2+. The main component of the beam has an energy of ~100 keV but there are secondary components with energies of ~34 and ~62 keV, which we don't understand. Figure 10 shows the N2+ ion beam profile, which is also about 4mm when the crystal-to-slit distance is 23 mm (the focal length). On the other hand the N2+ ion beam energy decreases by about 5% as the slit passes through the focal point of the beam as indicated by the count rate increase. Fig. 11. Ion beam energies at different slit positions and at different crystal-to-slit distances. Note that the maximum ion beam energy decrease occurs at the crystal focal length, which is 23mm. Finally Fig. 11 shows results of a study of the N2+ ion beam energy vs slit position for three different crystal-to-slit distances. In this case the pressure- sensitive energy-drop at 23 mm (the focal length) is nearly 10%. In conclusion self-focused energetic (<170 keV) electron beams are produced by heating to 1600C cylindrical LiNbO3 crystals with the - z base exposed in dilute gases at 2-8 mtorr on cooling. Conversely, positive gas-ion beams are produced by heating LiNbO3 crystals when the + z base is exposed again on cooling. Acknowledgments We are grateful to our colleagues, Tom Clegg, Bill Hooke, Sol Raboy, Brian Stoner and Eugen Merzbucher for insightful discussions. 0 25 50 75 100 125 150 175 200 225 250 0 500 1000 1500 2000 2500 3000 N2 Ion energy (keV) ~100 keV ~62 keV ~34 keV Counts 0 1 2 3 4 5 6 7 8 9 10 11 100 105 110 115 120 125 130 Slit position (mm) Ion beam energy (keV) 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 Ion count rate (c/sec) 0 1 2 3 4 5 6 7 8 9 10 11 12 100 105 110 115 120 125 Crystal d d = 23 mm d = 21 mm d = 80 mm 5 mm 9 mm 1 mm Ion beam energy (keV) Slit position (mm) a ) e-mail jdbjdb@binghamton.edu b)e-mail shafroth@physics.unc.edu 1. S. B. Lang, Sourcebook of Pyroelectricity, Gordon and Breach, New York 1974 2. J. D. Brownridge, Nature (London) 358,", + "= 23 mm d = 21 mm d = 80 mm 5 mm 9 mm 1 mm Ion beam energy (keV) Slit position (mm) a ) e-mail jdbjdb@binghamton.edu b)e-mail shafroth@physics.unc.edu 1. S. B. Lang, Sourcebook of Pyroelectricity, Gordon and Breach, New York 1974 2. J. D. Brownridge, Nature (London) 358, 278 (1992) 3. James Brownridge, and Sol Raboy, J. Appl. Phys. 86, 640 (1999) 4. S. M. Shafroth , W. Kruger and J.D. Brownridge, Nucl. Instr. and Meth., A 422, 1 (1999) 5. S. M. Shafroth and J. D. Brownridge, CP 475 American Institute of Physics , Applications of Accelerators in Research and Industry, Edited by J. L. Duggan and I. L. Morgan, p 1100 (1999) 6. J. D. Brownridge, S. M. Shafroth, D. Trott, B. Stoner, W. Hooke, Appl. Phys. Lett, 78, 1158 (2001) 7. J. D. Brownridge and S. M. Shafroth, Bull Am. Phys. Soc. 46, 106 & 164 (2001) 8. J. D. Brownridge and S. M. Shafroth, Appl. Phys. Lett. 79, 3364 (2001) 9. S. Bedair ,S. M. Shafroth and J. D. Brownridge, Bull. Am. Phys. Soc. 47, 65 (2002) 10 G. Rosenman, D. Shur, Ya. E. Krasik and A. Dunaevsky, J. Appl. Phys. 88, 6109 (2000). 11. B. Rosenblum, P. Brunlich, and J. P. Carrico, Appl. Phys. Lett. 25, 17(1974). 12. R. S. Weis and T. K. Gaylord, Appl. Phys. A 37, 191 (1985). 13. G. I. Rozenman, V. A. Bodyagin, Yu. L. Chepelev, and L. Ye. Isakova, Scripta Technica, ISSN in Radiotekhnika i elektronika, No9, (1987) pp. 1997- 1999. 14. D. Shur, G Rosenman, and Ya. E. Krasik, Appl. Phys. Lett. 70, 574 (1997). 15. Zdenek Sroubek, J. Appl. Phys. 88, 4452 (2000). 16. C. Winnewisser, P. Uhd Jepsen, M. Schall,V. V. Schyja and H. Helm, Appl. Phys. Lett. 70 3069 (1997). 17. J. D. Brownridge, Video clip, Dynamical Behavior of Beam as Pressure Increases, http://www.binghamton.edu/physics/bro wnridge.html 18. V. S. Kortov, A. F. Zatsepin, A. I. Gaprindashvili, M. M. Pinaeva, V. S. Vasil\"ev and I. A. Morozov, Sov. Phys. Tech. Phys. 25(9) 1126 (1981)", + "arXiv:physics/0002020v2 [physics.acc-ph] 10 Feb 2000 SIMULATION OF LASER-COMPTON COOLING OF ELECTRON BEAMS∗ TOMOMI OHGAKI Lawrence Berkeley National Laboratory Berkeley, California 94720, USA We study a method of laser-Compton cooling of electron beams. Using a Monte Carlo code, we evaluate the effects of the laser-electron interaction for transverse cooling. The optics with and without chromatic correction for the cooling are examined. The laser- Compton cooling for JLC/NLC at E0 = 2 GeV is considered. 1. Introduction A novel method of electron beam cooling for future linear colliders was proposed by V.Telnov.1 During head-on collisions with laser photons, the transverse distri- bution of electron beams remains almost unchanged and also the angular spread is almost constant. Because the Compton scattered photons follow the initial electron trajectory with a small additional spread due to much lower energy of photons (a few eV) than the energy of electrons (several GeV). The emittance ǫi = σiσ′ i re- mains almost unchanged (i = x, y). At the same time, the electron energy decreases from E0 to Ef. Thus the normalized emittances have decreased as follows ǫn = γǫ = ǫn0(Ef/E0) = ǫn0/C, (1) where ǫn0, ǫn are the initial and final normalized emittances, the factor of the emittance reduction C = E0/Ef. The method of electron beam cooling allows further reduction of the transverse emittances after damping rings or guns by 1-3 orders of magnitude.1 In this paper, we have evaluated the effects of the laser-Compton interaction for transverse cooling using the Monte Carlo code CAIN.2 The simulation calculates the effects of the nonlinear Compton scattering between the laser photons and the electrons during a multi-cooling stage. Next, we examine the optics for cooling with and without chromatic correction. The laser-Compton cooling for JLC/NLC3 at E0 = 2 GeV is considered in section 4. A summary of conclusion is given in section 5. ∗This work was supported in part by the U.S. Department of Energy under Contract No. DE- AC03-76SF00098. 1 Table 1. Parameters of the electron beams for laser-Compton cooling. The value in the parentheses is given by Telnov’s formulas. E0 (GeV) Ef (GeV) C ǫn,x/ǫn,y (m·rad) βx/βy (mm) σz (mm) δ (%) 2 0.2 10 7.4 × 10−8/2.9 × 10−8 4/4 0.5 11 (9.8) 5 1 5 3.0 × 10−6/3.0 × 10−6 0.1/0.1 0.2 19 (19) Table 2. Parameters of the laser beams for laser-Compton cooling. The value in the parentheses is given by Telnov’s formulas. E0 (GeV) λL (µm) x0 A (J) ξ RL,x/RL,y (mm) σL,z (mm) 2 0.5 0.076 300 (56) 2.1 (2.2) 0.3/0.3 1.25 5 0.5 0.19 20 (4) 1.5 (1.5) 0.1/0.1 0.4 2. Laser-Electron Interaction 2.1. Laser-Electron Interaction In this section, we describe the main parameters for laser-Compton cooling of electron beams. A laser photon of energy ωL (wavelength λL) is backward-Compton scattered by an electron beam of energy E0 in the interaction point (IP). The kinematics of Compton scattering is characterized by the dimensionless parameter1 x0 ≡4E0ωL m2ec4 = 0.019E0[GeV] λL[µm] , (2) where me is electron mass. The parameters of the electron and laser beams for laser-Compton cooling are", + "backward-Compton scattered by an electron beam of energy E0 in the interaction point (IP). The kinematics of Compton scattering is characterized by the dimensionless parameter1 x0 ≡4E0ωL m2ec4 = 0.019E0[GeV] λL[µm] , (2) where me is electron mass. The parameters of the electron and laser beams for laser-Compton cooling are listed in Tables 1 and 2. The parameters of the electron beam with 2 GeV are given for JLC/NLC case in section 4. The parameters of that with 5 GeV are used for simulation in the next subsection. The wavelength of laser is assumed to be 0.5 µm. The parameters of x0 with the electron energies 2 GeV and 5 GeV are 0.076 and 0.19, respectively. The required laser flush energy with ZR ≪lγ ≃le is1 A = 25le[mm]λL[µm] E0[GeV] (C −1) [J], (3) where ZR, lγ(∼2σL,z), and le(∼2σz) are the Rayleigh length of laser, and the bunch lengths of laser and electron beams. From this formula, the parameters of A with the electron energies 2 GeV and 5 GeV are 56 J and 4 J, respectively. The nonlinear parameter of laser field is1 ξ2 = 4.3 λ2 L[µm2] le[mm]E0[GeV](C −1). (4) In this study, for the electron energies 2 GeV and 5 GeV, the parameters of ξ are 2.2 and 1.5, respectively. The rms energy of the electron beam after Compton scattering is1 σe = 1 C2 \u0002 σ2 e0[GeV2] + 0.7x0(1 + 0.45ξ)(C −1)E 2 0 [GeV2] \u00031/2 [GeV], (5) where the rms energy of the initial beam is σe0 and the ratio of energy spread is defined as δ = σe/Ef. If the parameter ξ or x0 is larger, the energy spread after Compton scattering is increasing and it is the origin of the emittance growth in the defocusing optics, reacceleration linac, and focusing optics. The energy spreads δ for the electron energies 2 GeV and 5 GeV are 9.8% and 19%, respectively. The equilibrium emittances due to Compton scattering are1 ǫni,min = 7.2 × 10−10βi[mm] λL[µm] (i = x, y) [m · rad], (6) where βi are the beta functions at IP. From this formula we can see that small beta gives small emittance. However the large change of the beta functions between the magnet and the IP causes the emittance growth. Taking no account of the emittance growth, for the electron energies 2 GeV and 5 GeV, the equilibrium emittances are 5.8 × 10−9 m·rad and 1.4 × 10−10 m·rad, respectively. The equilibrium emittances depended on ξ in the case ξ2 ≫1 were calculated in Ref. 1. 2.2. Simulation of Laser-Electron Interaction For the simulation of laser-electron interaction, the electron beam is simply assumed to be a round beam in the case of E0 = 5 GeV and C = 5. Taking no account of the emittance growth of optics, the one stage for cooling consists two parts as follows: 1. The laser-Compton interaction between the electron and laser beams. 2. The reacceleration of electrons in linac. In the first part, we simulated the interactions by the CAIN code.2 This simulation calculates the effects of the nonlinear", + "growth of optics, the one stage for cooling consists two parts as follows: 1. The laser-Compton interaction between the electron and laser beams. 2. The reacceleration of electrons in linac. In the first part, we simulated the interactions by the CAIN code.2 This simulation calculates the effects of the nonlinear Compton scattering between the laser photons and the electrons. We assume that the laser pulse interacts with the electron bunch in head-on collisions. The βx and βy at the IP are fixed to be 0.1 mm. The initial energy spread of the electron beams is 1%. The energy of laser pulse is 20 J. The difference of the pulse energy between the simulation and the formula depends on the transverse sizes of the electron beams at IP. The polarization of the electron and laser beams are Pe=1.0 and PL=1.0 (circular polarization), respectively. When the x0 and ξ parameters are small, the spectrum of the scattered photons does not largely depend on the polarization combination. In order to accelerate the electron beams to 5 GeV for recovery of energy in the second part, we simply added the energy ∆E = 5 GeV −Eave for reacceleration, where Eave is the average energy of the scattered electron beams after the laser-Compton interaction. Here we define the transverse, longitudinal, and 6D emittances in the simulation. The x, y-transverse emittance is ǫn,i = q σ2 i σ2 i′ −(⟨i · i′⟩−⟨i⟩⟨i′⟩)2 (i = x, y), (7) where the symbol ⟨⟩means to take an average of all particles in a bunch. The longitudinal emittance is ǫn,l = q σ2zσ2pz −(⟨z · pz⟩−⟨z⟩⟨pz⟩)2/(mec). (8) The 6D emittance is defined as ǫ6N = ǫn,x · ǫn,y · ǫn,l. (9) Figure 1 shows the longitudinal distribution of the electrons after the first laser- Compton scattering. The average energy of the electron beams is 1.0 GeV and the energy spread δ is 0.19. The longitudinal distribution seems to be a boomerang. If we assume a short Rayleigh length of laser pulse, the energy loss of head and tail of beams is small. The number of the scattered photons per incoming particle and the average of the photon energy at the first stage are 40 and 96 MeV (rms energy 140 MeV), respectively. 0 1 2 E (GeV) ´10 1 0 0.6 0.4 0.2 0 1.0 0.8 Number of electrons × ×× × × × × × ×× × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×", + "× × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × ×× × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × ×× × × × × × × × × × ×× × × × × × × × × × × ×× × × × × × × × × × ×× × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × ×", + "× × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × ×× × × × × ×× × × × × × × × × × × × × × × × ×", + "× × × × × × × × × ×× × × × × × × × × × × × × × × × × × × ×× × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × ×× × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × ×× × × × ×× × × × × × × × × × × × × × ×× ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×", + "× × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × 1.0 -1.0 0 0.5 -0.5 Z (mm) E (GeV) 1.5 1.0 0.5 0 2.0 (a) (b) Fig. 1. The longitudinal distribution of the electrons. (a) The energy vs. z. (b) The energy distribution of the electrons. The bin size is 40 MeV. 0 1 2 3 4 5 6 7 8 9 Stage 10 -8 10 -7 Transverse size (m) ( ) si i=x,y sy sx Fig. 2. The transverse sizes of the electron beams. 0 1 2 3 4 5 6 7 8 9 Stage Angle (rad) ( ) qi i=x,y 10 -3 10 -5 10 -4 qy qx Fig. 3. The angles of the electron beams. The transverse sizes of the electron beams in the multi-stage cooling are shown in Fig. 2. During collisions with the laser photons, the transverse distribution of the electrons remains almost unchanged. But they decrease when we focus them for the next laser-Compton interaction due to the lower normalized emittance and the fixed β-function at IP (σi = p βiǫn,i/γ). The angles of the electron beams in the multi-stage cooling are shown in Fig. 3. As a result of reacceleration, the angles of the electrons decrease. They increase when we focus them for the next laser- Compton interaction. Finally the angles attain the average of Compton scattering angle and the effect of cooling saturates. Figure 4 shows the transverse emittances of the electron beams in the multi- stage cooling. From Eq.(6), ǫni,min = 1.4×10−10 m·rad, and the simulation presents ǫni,min = 1.2×10−9 m·rad. Figure 5 shows the longitudinal emittance of the electron beams in the multi-stage cooling. Due to the increase of the energy spread of the electron beams from 1% to 19%, the longitudinal emittance rapidly", + "stage cooling. From Eq.(6), ǫni,min = 1.4×10−10 m·rad, and the simulation presents ǫni,min = 1.2×10−9 m·rad. Figure 5 shows the longitudinal emittance of the electron beams in the multi-stage cooling. Due to the increase of the energy spread of the electron beams from 1% to 19%, the longitudinal emittance rapidly increases at the first stage. After the first stage, the normalized longitudinal emittance is stable. The 6D emittance of the electron beams in the multi-stage cooling is shown in Fig. 6. The second cooling stage has the largest reduction for cooling. The 8th or 9th cooling stages have small reduction for cooling. The initial and final 6D emittances ǫ6N are 1.5 × 10−13 (m·rad)3 and 1.2 × 10−19 (m·rad)3, respectively. Figure 7 shows the polarization of the electron beams in the multi-stage cooling. The decrease of the polarization during the first stage is about 0.06. The final polarization Pe after the multi-stage cooling is 0.54. 0 1 2 3 4 5 6 7 8 9 Stage 10 -5 10 -9 10 -8 10 -10 10 -7 10 -6 Emittance (m rad) ( ) e × n i, i=x,y en x, en y, Fig. 4. The transverse emittances of the electron beams. 0 1 2 3 4 5 6 7 8 9 Stage Emittance (m rad) e × n l, 0.1 0.09 0.08 0.07 0.06 0.05 0.01 0.04 0.03 0.02 Fig. 5. The longitudinal emittance of the electron beams. 3. Optics Design for Laser-Compton Cooling 3.1. Optics without chromaticity correction There are three optical devices for the laser-Compton cooling of electron beams as follows: 1. The focus optics to the first IP. 0 1 2 3 4 5 6 7 8 9 Stage 10 -18 10 -16 10 -20 10 -14 10 -12 Emittance (m rad) e × 6 N 3 Fig. 6. The 6D emittance of the electron beams. 0 1 2 3 4 5 6 7 8 9 Stage Polarization 1.0 0.8 0.6 0 0.4 0.2 Fig. 7. The polarization of the electron beams. 2. The defocus optics from the first IP to the reacceleration linac. 3. The focus optics from the linac to the next IP. Figure 8 shows schematics of the laser-Compton cooling of electron beams. The optics 1 is focusing the electron beams from a few meters of β-function to several millimeters in order to effectively interact them with the laser beams. The optics 2 is defocusing them from several millimeters to a few meters for reacceleration of electron beams in linac. In a multi-stage cooling system, the optics 3 is needed for cooling in the next stage. The problem for the focus and defocus optical devices is the energy spread of electrons and the electron beams with a large energy spread are necessary to minimize or correct the chromatic aberrations avoiding emittance growth. Linac Defocus lens Focus lens Focus lens IP IP 5 GeV 1GeV 5 GeV d=4% d=1% d=19% Laser Beam Next Stage Fig. 8. Schematics of the laser-Compton cooling of electron beams. In this subsection, we discuss the optics for laser-Compton cooling without chro- matic", + "the chromatic aberrations avoiding emittance growth. Linac Defocus lens Focus lens Focus lens IP IP 5 GeV 1GeV 5 GeV d=4% d=1% d=19% Laser Beam Next Stage Fig. 8. Schematics of the laser-Compton cooling of electron beams. In this subsection, we discuss the optics for laser-Compton cooling without chro- matic corrections. For the focus and defocus of the beams, we use the final doublet system which is similar to that of the final focus system of the future linear col- liders.3 The pole-tip field of the final quadrupole BT is limited to 1.2 T and the pole-tip radius a is greater than 3 mm. The strength of the final quadrupole is κ = BT /(aBρ) ≤120/E[GeV] [m−2], (10) where B, ρ, and E are the magnetic field, the radius of curvature and the energy of the electron beams. In our case, the electron energies in the optics 1, 2, and 3 are 5.0, 1.0, and 5.0 GeV, respectively and the limit of the strength of the quadrupole in laser cooling is much larger than that of the final quadrupole of the future linear colliders. Due to the low energy beams in laser cooling, the synchrotron radiation from quadrupoles and bends is negligible. The difference of three optical devices is the amount of the energy spread of the beams. In the optics 1,2, and 3, the beams have one, several tens, and a few % energy spread. In order to minimize the chromatic aberrations, we need to shorten the length between the final quadrupole and the IP. In this study, the length from the face of the final quadrupole to the IP, l is assumed to be 2 cm. Here we estimated the emittance growth in the optics 2, because the chromatic effect in the optics 2 is the most serious. Figure 9 shows the defocus optics without chromaticity correction for laser-Compton cooling by the MAD code.4 The input file is attached to Ref. 5. The parameters of the electron beam for laser-Compton cooling at E0 = 5 GeV and C = 5 are listed in Table 3. The initial βx and βy after laser-Compton interaction are 20 mm and 4 mm, respectively. The final βx and βy are assumed to be 2 m and 1 m, respectively. The initial and final αx(y) with no energy spread δ = 0 are 0 in this optics. The strength κ of the final quadrupole for the beam energy of 1 GeV from Eq. (10) is assumed to be 120 m−2. Table 3. Parameters of the electron beam for laser-Compton cooling at E0 = 5 GeV and C = 5 for the optics design. E0 (GeV) ǫn,x/ǫn,y (m·rad) βx/βy (mm) σx/σy (m) σz (mm) 5 1.06 × 10−6/1.6 × 10−8 20/4 3.3 × 10−5/1.8 × 10−7 0.2 In our case, the chromatic functions ξx and ξy are 18 and 148, respectively. The momentum dependence of the emittances in the defocus optics without chromaticity correction is shown in Fig. 10. In the paper,6 the analytical study by thin-lens approximation has been studied for the focusing system, and", + "0.2 In our case, the chromatic functions ξx and ξy are 18 and 148, respectively. The momentum dependence of the emittances in the defocus optics without chromaticity correction is shown in Fig. 10. In the paper,6 the analytical study by thin-lens approximation has been studied for the focusing system, and here the transverse emittances are calculated by a particle-tracking simulation. The 10000 particles are tracked for the transverse and longitudinal Gaussian distribution by the MAD code. The relative energy spread δ is changed from 0 to 0.4. Due to the larger chromaticity ξy, the emittance ǫy is rapidly increasing with the energy spread δ. If we set a limit of 200% for ∆ǫi/ǫi (i = x, y), the permissible energy spread δx and δy are 0.11 and 0.012 which mean the momentum band widths ±22% and ±2.4%, respectively. The results are not sufficient for cooling at E0 = 5 GeV and C = 5, because the beams through the defocusing optics have the energy spread of several tens %. On the one hand, the optics can be useful as the optics 1 and 3 with the energy spread of a few %. 3.2. Optics with chromaticity correction The optics without chromaticity correction for the optics 2 does not work as we saw in the previous subsection. In this subsection we apply the chromaticity correction for the optics 2. The lattice for cooling is designed referring to the final focus system of the future linear colliders by K. Oide.7 The final doublet system 0.0 .25 .50 .75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 s (m) 0.0 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. b (m ) 0.0 20. 40. 60. 80. 100. 120. 140. 160. W b x b y Wx Wy Fig. 9. The defocus optics without chromaticity correction for laser-Compton cooling. 0 0.1 0.4 0.2 0.3 d Emittance (m rad) e × y 10 -8 10 -9 10 -12 10 -11 10 -10 Emittance (m rad) e × x 10 -8 10 -9 10 -10 0 0.1 0.4 0.2 0.3 d Fig. 10. Momentum dependence of the emittances in the defocus optics without chromaticity correction. is the same lattice as the optics before subsection. The method of chromaticity correction uses one family of sextupole to correct for vertical chromaticity and moreover we added two weak sextupoles in the lattice to correct for horizontal chromaticity. Figure 11 shows the defocus optics with chromaticity correction for the laser-Compton cooling. The input file is attached to Ref. 5. The total length of the lattice is about 63 m. The momentum dependence of the emittances in the defocus optics with chro- maticity correction is shown in Fig. 12. The 10000 particles are tracked for the transverse and longitudinal Gaussian distribution by the MAD code. The relative energy spread δ is changed from 0 to 0.06 with the conservation κ2 θB, where κ2 and θB are the strength of the sextupole and the angle of the bending magnet. The initial βx and βy after laser-Compton interaction are 20 mm and", + "distribution by the MAD code. The relative energy spread δ is changed from 0 to 0.06 with the conservation κ2 θB, where κ2 and θB are the strength of the sextupole and the angle of the bending magnet. The initial βx and βy after laser-Compton interaction are 20 mm and 4 mm, respectively. The final βx and βy are assumed to be 2 m and 1 m, respectively. The initial and final αx(y) with no energy spread δ = 0 are 0 in this optics. After the chromaticity correction, the chromaticity functions ξx and ξy are 9.3 and 1.6, respectively. If we set a limit of 200% for ∆ǫi/ǫi(i = x, y), the permissible energy spread δx and δy are 0.040 and 0.023 which mean the momentum band widths ±8% and ±4.6%, respectively. By the comparison with the results of the optics without chromaticity correction at a limit of 200% for ∆ǫi/ǫi(i = x, y), the ǫy of the optics with chro- maticity correction is about two times larger than that of the one before subsection, but the ǫx of the optics with chromaticity correction is three times smaller than that of the one before. The results are still not sufficient for cooling with E0 = 5 GeV and C = 5. These results emphasize the need to pursue further ideas for plasma lens.8 d E/ p 0c = 0 0.0 10. 20. 30. 40. 50. 60. 70. 80. s (m) 0.0 100. 200. 300. 400. 500. 600. b (m) 0.0 .0005 .0010 .0015 .0020 .0025 .0030 .0035 .0040 .0045 .0050 D (m) b x b y Dx Dy . s (m) d E/ p 0c = 0 . 0.0 20. 40. 60. 80. 100. 120. 140. 160. 180. 200. W 0.0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 m (rad/ 2p) Wx Wy m x m y 0.0 10. 20. 30. 40. 50. 60. 70. 80. Fig. 11. The defocus optics without chromaticity correction for laser-Compton cooling. Emittance (m rad) e × y 10 -9 10 -12 10 -11 10 -10 0 0.02 0.06 0.04 d Emittance (m rad) e × x 10 -8 10 -9 10 -10 0 0.02 0.06 0.04 d Fig. 12. Momentum dependence of the emittances in the defocus optics with chromaticity correction. 4. Laser-Compton Cooling for JLC/NLC at E0 = 2 GeV 4.1. Optics For the future linear colliders, the method of laser-Compton cooling is effective to reduce the transverse emittances after damping rings. Where can it be placed? There are two possibilities for JLC/NLC9 as follows: 1. After the first bunch compressor (BC1) and before the pre-linac. E0 = 2 GeV and σz = 0.5 mm. 2. After the second bunch compressor (BC2) and before the main linac. E0 = 10 GeV and σz = 0.1 mm. Case 2 needs a large energy for recovery after Compton scattering and we consider case 1 in this study. The parameters of the electron and laser beams for laser- Compton cooling for JLC/NLC at E0 = 2 GeV and C = 10 are listed in Tables 1", + "= 0.1 mm. Case 2 needs a large energy for recovery after Compton scattering and we consider case 1 in this study. The parameters of the electron and laser beams for laser- Compton cooling for JLC/NLC at E0 = 2 GeV and C = 10 are listed in Tables 1 and 2. The energy of laser pulse is 300 J. The simulation results of the laser-electron interaction by the CAIN code are summarized as follows. The energy spread of the electron beam is 11%. The decrease of the longitudinal polarization of the electron beam is 0.038 (Pe = 1.0, PL = 1.0). The number of the scattered photons per incoming particle and the average of the photon energy are 200 and 8.9 MeV (rms energy 19 MeV), respectively. Table 4. Parameters of the defocus optics for laser-Compton cooling for JLC/NLC at E0=2 GeV and C = 10. l Length of Q1 Field of Q1 Aperture Total length 5 mm 2 cm 1.2 Tesla 0.5 mm 7.4 cm The electron energy after Compton scattering in case 2 is 0.2 GeV and the strength of the final quadrupole from Eq. (10) is 600 m−2. Table 4 lists the param- eters of the defocusing optics for laser-Compton cooling for JLC/NLC at E0 = 2 GeV and C = 10. The final βx and βy are assumed to be 1 m and 0.25 m, respec- tively. The chromaticity functions ξx and ξy are 18 and 23, respectively. Using the MAD code, the emittance growth in the defocus optics are ∆ǫdefocus n,x = ǫn,x −ǫn,x0 ∼1.0ǫn,x0 ∼7.6 × 10−8 [m · rad], (11) ∆ǫdefocus n,y = ǫn,y −ǫn,y0 ∼1.6ǫn,y0 ∼4.6 × 10−8 [m · rad], (12) where the normalized emittances before and after the defocus optics are ǫn,i0 and ǫn,i (i = x, y), respectively. The emittance growth in the other two focus optics are negligible. 4.2. Reacceleration Linac In the reacceleration linac, there are two major sources of the emittance increase9 as follows: 1. The emittance growth due to the misalignment of the quadrupole magnet and the energy spread. 2. The emittance growth due to the cavity misalignment. The emittance growth due to these sources in the reacceleration linac (L-band linac) are formulated by K. Yokoya9 ∆ǫlinac n,x ∼3.4 × 10−9 [m · rad] ∼0.045ǫn,x0, (13) ∆ǫlinac n,y ∼3.4 × 10−9 [m · rad] ∼0.12ǫn,y0. (14) The final emittance growth and the final emittance with C = 10 are ∆ǫn,x ∼7.9 × 10−8 [m · rad] ∼1.1ǫn,x0 ⇒ǫn,x ∼0.21ǫn,x0, (15) ∆ǫn,y ∼4.9 × 10−8 [m · rad] ∼1.7ǫn,y0 ⇒ǫn,y ∼0.27ǫn,y0. (16) The total reduction factor of the 4D transverse emittance of the laser-Compton cooling for JLC/NLC at E0 = 2 GeV is about 18. The decrease of the polarization of the electron beam is 0.038 due to the laser-Compton interaction. 5. Summary We have studied the method of laser-Compton cooling of electron beams. The ef- fects of the laser-Compton interaction for cooling have been evaluated by the Monte Carlo simulation. From the simulation in the multi-stage cooling, we presented that the low emittance beams with ǫ6N", + "to the laser-Compton interaction. 5. Summary We have studied the method of laser-Compton cooling of electron beams. The ef- fects of the laser-Compton interaction for cooling have been evaluated by the Monte Carlo simulation. From the simulation in the multi-stage cooling, we presented that the low emittance beams with ǫ6N = 1.2 × 10−19(m·rad)3 can be achieved in our beam parameters. We also examined the optics with and without chromatic correc- tion for cooling, but the optics are not sufficient for cooling due to the large energy spread of the electron beams. The laser-Compton cooling for JLC/NLC at E0 = 2 GeV and C = 10 was considered. The total reduction factor of the 4D transverse emittance of the laser- Compton cooling is about 18. The decrease of the polarization of the electron beam is 0.038 due to the laser-Compton interaction. Acknowledgments We would like to thank Y. Nosochkov, K. Oide, T. Takahashi, V. Telnov, M. Xie, and K. Yokoya for useful comments and discussions. References 1. V. Telnov, Phys. Rev. Lett. 78, 4757 (1997); ibid. 80, 2747 (1998); in Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, Monterey, CA, Jan 4-9, 1998, BUDKERINP-98-33 (1998). 2. P. Chen, T. Ohgaki, A. Spitkovsky, T. Takahashi, and K. Yokoya, Nucl. Instrum. and Methods Phys. Res. A 397, 458 (1997). 3. Zeroth-Order Design Report for the Next Linear Collider, LBNL-PUB-5424, SLAC- Report-474 (1996); JLC Design Study, KEK-Report-97-1 (1997); Conceptual Design of a 500 GeV Electron Positron Linear Collider with Integrated X-Ray Laser Facility, DESY-97-048, ECFA-97-182 (1997). 4. H. Grote and F.C. Iselin, The MAD Program (Methodical Accelerator Design) Version 8.19: User’s Reference Manual, CERN-SL-90-13-AP (1996). 5. T. Ohgaki, LBNL-44380 (1999). 6. B.W. Montague and F. Ruggiero, CLIC-NOTE-37 (1987). 7. K. Oide, Nucl. Instrum. Meth. Phys. Res. A 276, 427 (1989); in Proceedings of the DPF Summer Study on High Energy Physics in the 1990’s, Snowmass, CO, Jun 27-Jul 15, 1988, SLAC-PUB-4806 (1988); in Proceedings of the 1st Workshop on the Japan Linear Collider (JLC), Tsukuba, Japan, Oct 24-25, 1989, KEK-Preprint-89-190 (1989). 8. P. Chen, K. Oide, A.M. Sessler, and S.S. Yu, Phys. Rev. Lett. 64, 1231 (1990); in Proceedings of the Fourteenth International Conference on High Energy Accelerators, Tsukuba, Aug 22-26, 1989, SLAC-PUB-5060 (1989). 9. K. Yokoya, in Proceedings of the International Symposium on New Visions in Laser- Beam Interactions, Tokyo, Oct 11-15, 1999.", + "arXiv:cond-mat/0507150v1 [cond-mat.str-el] 6 Jul 2005 Electronic reconstruction in correlated electron heterostructures Satoshi Okamoto and Andrew J. Millis Department of Physics, Columbia University, 538 West 120th Street, New York, NY 10027 ABSTRACT Electronic phase behavior in correlated-electron systems is a fundamental problem of condensed matter physics. We argue here that the change in the phase behavior near the surface and interface, i.e., electronic reconstruction, is the fundamental issue of the correlated-electron surface or interface science. Beyond its importance to basic science, understanding of this behavior is crucial for potential devices exploiting the novel properties of the correlated systems. We present a general overview of the field, and then illustrate the general concepts by theoretical studies of the model heterostructures comprised of a Mott-insulator and a band-insulator, which show that spin (and orbital) orderings in thin heterostructures are generically different from the bulk and that the interface region, about three-unit-cell wide, is always metallic, demonstrating that electronic reconstruction generally occurs. Predictions for photoemission experiments are made to show how the electronic properties change as a function of position, and the magnetic phase diagram is determined as a function of temperature, number of layers, and interaction strength. Future directions for research are also discussed. Keywords: Correlated-electron systems, Mott insulator, band insulator, heterostructure, interface, magnetism 1. INTRODUCTION Correlated electron systems such as transition metal oxides are materials in which strong electron-electron or electron-lattice interactions produces behavior incompatible with the standard density functional plus Migdal- Eliashberg theory which describes most compounds. The past decade has seen tremendous progress in the physics and materials science of correlated-electron systems. Significant improvements in crystal and film growth, in measurement techniques and in theory have led to a much improved understanding of the bulk properties of these materials. An important finding is that correlated electron systems exhibit a multiplicity of interesting phases (superconducting, magnetic, charge, and orbitally ordered) along with associated excitations. For recent reviews, see Ref. 1, or the articles in Ref. 2. The recent success in treating bulk properties suggests that the time is ripe for a systematic study of the surface and interface properties of correlated electron systems. In addition to its basic importance as a fundamental question in materials science, correlated electron surface/interface science should provide the necessary scientific background for study of potential devices exploiting correlated electron properties, because essentially any device must be coupled to the rest of the world via motion of electrons through an interface. The fundamental interest of bulk correlated electron materials lies in novel phases they exhibit, and we therefore suggest that the fundamental issue for the nascent field of correlated electron surface science is how does the electronic phase at the surface or interface differ from that in the bulk; in other words, what is the electronic surface reconstruction. As in the ordinary surface or interface science, many physics and material science issues arise in considering the behavior of correlated electrons near surfaces and interfaces. Atomic reconstruction may occur, and may change the underlying electronic structure. For example, the authors of Ref. 3 argue that, in two-dimensional ruthenates, a change", + "in the ordinary surface or interface science, many physics and material science issues arise in considering the behavior of correlated electrons near surfaces and interfaces. Atomic reconstruction may occur, and may change the underlying electronic structure. For example, the authors of Ref. 3 argue that, in two-dimensional ruthenates, a change in tilt angle of the surface RuO6 octahedra increases the electronic hopping, thereby allowing the metallic phase to persist to lower T than in the bulk. Reduced coordination number at surfaces is supposed to enhance correlation effects as discussed by Potthoffand Nolting,4, 5 Schwieger et al.,6 and Liebsch.7 Also, Hesper and co-workers have shown that the [111] surface of K3C60 differs from the bulk. They noted that a change in structure will lead to changes in Madelung potentials, and to the screening which helps define the Further author information: (Send correspondence to S.O.) E-mail: okapon@phys.columbia.edu, Telephone: +1-212- 854-6712 values of many-body interaction parameters.8 Leakage of charge across an interface may change densities away from the commensurate values required for insulating behavior. On the experimental side, a variety of heterostructures have been fabricated and studied. For the surface, i.e., interface between a material and a vacuum, Maiti and collaborators have shown that the surface and bulk electronic phases of Ca1−xSrxVO3 are significantly different.9 They argued that the enhanced incoherent part of the photoemission spectra may be due to the reduced coordination number at the surface and/or surface recon- struction. Moore and collaborators have demonstrated that in the correlated electron system Ca1.9Sr0.1RuO4 (which exhibits a Mott metal-insulator transition), the surface layers remain metallic down to a lower tempera- ture than does the bulk system.3 We have proposed that in this system electron-lattice coupling is crucial for the Mott metal-insulator transition.10 Therefore, the surface Mott transition in these materials may be different from that in the bulk; at the surface, due to the larger spring constant (estimated from the higher phonon frequency), electron-lattice coupling is reduced, and surface metal transition is suppressed. A variety of interfaces between different materials includes high-Tc cuprates,11–13 Mott-insulator and band- insulator heterostructure,14 and superlattices of transition-metal oxides.15–21 Izumi and co-workers have fab- ricated digital heterostructures composed of different transition metal oxides and have demonstrated changes in electronic phase and other properties depending on the thicknesses of different layers.15 Warusawithana and collaborators fabricated and measured a variety of superlattices of dielectric materials with broken inversion symmetry.19 In an experimental tour-de-force, Ohtomo and co-workers have demonstrated the fabrication of atomically precise digital heterostructures involving a controllable number n of planes of LaTiO3 (a correlated- electron Mott-insulating material) separated by a controllable number m of planes of SrTiO3 (a more conventional band-insulating material) and have measured both the variation of electron density transverse to the planes and the dc transport properties of the heterostructure.14 Their work opens the door to controlled studies both of correlated electron physics in confined dimensions and of the behavior of the interface between a correlated sys- tem and a uncorrelated one. Bowen and collaborators fabricated mangnites based tunneling magnetoresistance (TMR) junctions, and succeeded in obtaining the high spin", + "heterostructure.14 Their work opens the door to controlled studies both of correlated electron physics in confined dimensions and of the behavior of the interface between a correlated sys- tem and a uncorrelated one. Bowen and collaborators fabricated mangnites based tunneling magnetoresistance (TMR) junctions, and succeeded in obtaining the high spin polarization as in the bulk materials at the lowest temperature.20 However, magnetoresistance is lost well below the bulk Curie temperature. Similarly, Nakagawa and collaborators fabricated manganite-titanate heterojunctions and measured their current-bias voltage char- acteristics and junction capacitance with and without the applied magnetic field.21 They observed a significant magnetic-field dependence of the junction capacitance at low temperature indicating the change in the electronic state (possibly spin and orbital states) of manganites region near the interface under the magnetic field. Finally, with an elaborate sample preparation, Schneider et al. were successful in extrapolating the resistance of single grain boundary of YBa2Cu3O7−δ.13 This is clearly supported by the fact that the resistance is not affected by the onset of the superconducting transition. The resistance is found to decrease linearly with the increase of temperature, and it has been suggested that randomly distributed magnetic impurities are responsible for the scattering mechanism.22 On the theoretical side, there have been several developments. The enhanced correlations near the surface due to the reduced coordination number4–7 could presumably induce surface magnetic ordering, this had been discussed in a mean field treatment of the Hubbard model by Potthoffand Nolting.23 Matzdorf et al. proposed that ferromagnetic ordering is stabilized at the surface of two-dimensional ruthenates by a lattice distortion,24 but this has not yet been observed. Calderon et al. discussed possible surface antiferromagnetism in manganites arising from a competition between reduced kinetic energy and antiferromagnetic superexchange interaction between the nearest-neighboring local spins.25 The effect of bulk strain on the magnetic ordering in perovskite manganites was discussed by Fang et al.26 Effect of strain had been intensively studied on the ferroelectric materials by using the first-principle calculation. (For example, see Refs. 27 and 28.) Further, Ederer and Spaldin studied the effects of strain and oxygen vacancy on multiferroicity in bismuth ferrite.29 Popovic and Satpathy applied LSDA and LSDA+U methods to compute the magnetic properties of LaTiO3/SrTiO3 heterostructures fabricated by Ohtomo et al.30 Going beyond the study of static properties, Freericks applied the dynamical-mean- field method to the correlated [001] heterostructures comprised of non-correlated and strongly-correlated regions, and computed the conductance perpendicular to the plane.31 In his model, the correlated region is described by the Falicov-Kimball model, which is a simplified version of the Hubbard model neglecting the electron hopping integral of one of two spin components, and the computations are limited to the particle-hole symmetric case (uniform charge density) for simplicity. Extension to the general model, and in particular, to the situation where the charge density is spatially modulated are important future directions. Sorting out the different contributions and assessing their effect on the many-body physics is a formidable task, which will require a sustained experimental and theoretical effort. The experiment of Ohtomo et al. offers an attractive starting point. In this", + "where the charge density is spatially modulated are important future directions. Sorting out the different contributions and assessing their effect on the many-body physics is a formidable task, which will require a sustained experimental and theoretical effort. The experiment of Ohtomo et al. offers an attractive starting point. In this system, the near lattice match (1.5 % difference in lattice parameter) and chemical similarity of the two components (LaTiO3 and SrTiO3) suggests that atomic reconstructions, strain, and renormalizations of many-body parameters are of lesser importance, so the physical effects of electronic reconstruction can be isolated and studied. Furthermore, the near Fermi surface states are derived mainly from the Ti d orbitals,32 and correspond to narrow bands well described by tight-binding theory. Therefore, the model calculation of heterostructures of the type created by Ohtomo et al. is expected to be a good starting point toward a general understanding of the correlated electron surface and interface problem. In the rest of this paper we review our work on theoretical analysis of the correlated electron behavior to be expected in lattice-matched digital heterostructures of the type created by Ohtomo et al.14 We focus on electrons in the Ti-derived d-bands and include the effects of the long-ranged electric fields arising both from the La atoms and the electronic charge distribution. Ti d-bands are represented by either a realistic t2g three- band Hubbard model or a simplified single-band Hubbard model. The Hartree-Fock (HF) approximation is applied to treat the on-site interactions of the three-band model. We calculate the electronic phase diagram as a function of on-site interaction parameter and number of La layers and for the relevant phases determine the spatial variation of charge, spin and orbital densities.33, 34 We obtain a complex set of behaviors depending on interaction strength and number of La layers. Generally, we find a crossover length of approximately three unit cells, so that units of six or more LaTiO3 layers have a central region which exhibits bulk-like behavior. The outermost ∼3 layers on each side are however metallic (in contrast to the insulating behavior of bulk LaTiO3 and SrTiO3). For very thin superlattices the ordering patterns differ from that in bulk. While the HF approximation provides the correct tendency towards magnetically and orbitally ordered states, this method is known to be an inadequate representation of strongly correlated materials, and in particular, does not include the physics associated with proximity to the Mott insulating state. Therefore, as a step to go beyond HF approximation, we apply the dynamical-mean-field method,35 which provides a much better representation of the electronic dynamics associated with strong correlations, to a simplified single-band model, and investigate the dynamical properties of correlated-electrons.36,37 The dynamical-mean-field analysis confirms the important results obtained by the HF analysis , i.e., different phases in thin heterostructures than in the bulk and metallic edge, but provides significant improvement over the HF results for the emergence of the metallic behavior, the magnetic transition temperatures and order parameter distributions. The rest of this paper is organized as follows: In Sec. 2, we present our theoretical models and methods", + "heterostructures than in the bulk and metallic edge, but provides significant improvement over the HF results for the emergence of the metallic behavior, the magnetic transition temperatures and order parameter distributions. The rest of this paper is organized as follows: In Sec. 2, we present our theoretical models and methods applied, Sec. 3 presents results of the HF analysis of the realistic three-band model, and Sec. 4 presents results of the dynamical-mean-field analysis of the single-band model. Section 5 is devoted to conclusion and discussion. Part of the results presented in this paper has already been published in Refs. 33, 34, and 36, and can be seen in Ref. 37. 2. MODEL AND METHOD In this study, we consider an infinite crystal of SrTiO3, into which n adjacent [001] planes of LaTiO3 have been inserted perpendicular to one of the Ti-Ti bond directions, as shown in Fig. 1 ([001] heterostructure). We choose the z direction to be perpendicular to the LaTiO3 planes, so the system has (discrete) translation symmetry in the xy direction. Both LaTiO3 and SrTiO3 crystallize in the simple ABO3 perovskite structure38, 39 (more precisely, very small distortions occur) and as noted by Ref. 14 the lattice constants of the two materials are almost identical; aLaT iO3 ≃aSrT iO3 = 3.9 ˚A, minimizing structural discontinuities at the interface and presumably aiding in the growth of high quality digital heterostructures. Therefore, in this study, we neglect lattice distortions and focus on the purely electronic model. Possible effects of lattice distortions will be briefly discussed below. We consider the following two model heterostructures: (1) realistic three-band model and (2) single-band model. We apply the HF approximation to the three-band model and discuss static properties such as spin and /D 7L 6U Q \u0003\u0015 Figure 1. Schematic figure of the model used in the present study. Open and crossed circles show the positions of Sr and La ions. (La layer number n = 2) The dots show the positions of Ti ions. x, y axes are chosen to be parallel to the La plane, and z axis is perpendicular to the plane. orbital orderings of the heterostructures. The single-band heterostructure is not a fully realistic representation of the systems studied Ref. 14, but the essential physics associated with charge leakage and strong correlation is included. The single-band model is analyzed using the dynamical-mean-field theory (DMFT). Applying beyond-HF methods including the DMFT to the multi-band model is highly desirable, but this is beyond the current computational ability. In order to apply the DMFT to the heterostructure problem, it is required to solve many quantum impurity models coupled with each other self-consistently. Further, numerically expensive method is usually applied to solve the impurity model such as the quantum Monte-Carlo (QMC) method or the exact-diagonalization (ED) method. However, as will be discussed later, interesting physics appears at strong coupling regime, the interaction strength is larger than the band width, and at low-but-non- zero temperature, lower than the magnetic ordering temperature of the bulk material. This regime is not easily accessible either by QMC and by ED.", + "method. However, as will be discussed later, interesting physics appears at strong coupling regime, the interaction strength is larger than the band width, and at low-but-non- zero temperature, lower than the magnetic ordering temperature of the bulk material. This regime is not easily accessible either by QMC and by ED. In order to apply the DMFT method to realistic multi-band models, numerically inexpensive but reliable impurity solvers are required. Work in this direction is still in progress. In both the models, the hamiltonian Htot is comprised of three terms: Htot = Hhop + Hon−site + HCoul with electron hopping term Hhop, on-site interaction term Hon−site, and long-ranged Coulomb interaction term HCoul. The hopping term Hhop and the on-site interaction term Hon−site vary depending on the model one considers as presented in the following subsections, while the long-ranged Coulomb interaction term HCoul has the same form and consists of two terms: (i) the Coulomb force arising from the extra charge on the La relative to Sr defining the heterostructure, and (ii) the Coulomb force arising from the electrons on other Ti sites. Thus, HCoul = X i V (⃗ri)ntot(⃗ri) (1) with V (⃗ri) = − X La-sites j e2 ε|⃗rLa j −⃗ri| + 1 2 X Ti-sites j̸=i e2ntot(⃗rj) ε|⃗rj −⃗ri| (2) and ntot(⃗ri) = P σ(α) d† i(α)σdi(α)σ (sum over the orbital index α is necessary for the three-band model). Here the ⃗ri are the positions of the Ti(B) sites in the ABO3 lattice and ⃗rLa j label the actual positions of the La ions, which reside on the A sites. We denote the dielectric function of the host lattice by ε. An interesting issue arises here: SrTiO3 is a nearly ferroelectric material.40 The static dielectric constant becomes very large at long wavelength and low temperatures, but ε is much smaller at high frequencies, room temperature, or short length scales. Also the polarization P will increase more slowly at higher fields, and relevant quantity is P/E. In our work, we have chosen ε ∼15 as a compromise between these effects, and have chosen a dimensionless parameter for the long- ranged Coulomb interaction Ec = e2/(εat) = 0.8 corresponding to a lattice constant a = 3.9 ˚A transfer intensity t = 0.3 eV. The effect of different choices of ε will be discussed later. In addition to the strong ferroelectric tendency, other issues have been neglected in theoretical work so far. We emphasize that incorporating these effects in a more realistic manner is a important question for future research. Important effects include (I) Change in the electron transfer intensity: As can be seen in the TEM image of Fig. 1 in Ref. 14, Ti sites do not show significant displacement throughout the heterostructure. Thus, change in the transfer intensity between neighboring Ti d-orbitals δt is given by δt ∝δ2 with δ the oxygen displacement. Therefore, it is expected be small. However, when a more realistic model including O p-orbitals is considered, this effect can not be neglected because the Ti–O transfer is changed linearly in δ. Further, (II) Change in the on-site interaction", + "δt is given by δt ∝δ2 with δ the oxygen displacement. Therefore, it is expected be small. However, when a more realistic model including O p-orbitals is considered, this effect can not be neglected because the Ti–O transfer is changed linearly in δ. Further, (II) Change in the on-site interaction parameters: Due to the lattice distortion, screening effect might be different, thus, changing the interaction parameters. (III) Degeneracy lifting in the Ti d-level: Due to the small but non-zero mismatch in the lattice constants, a = 3.91 ˚A for SrTiO3 substrate, and a = 3.97 ˚A for LaTiO3,39 The Ti d-orbitals in LaTiO3 region would suffer from an in-plane compressive strain, and the energy levels of the orbitals elongated in the plane would be raised. (IV) Absence of inversion symmetry: This would admix the Ti 3d t2g-orbital with the orbitals with the different symmetry including the Ti 3d eg- and 4p-orbitals via Ti–O hybridization. (V) Chemical effect: Because of the chemical difference between the two compounds, there may be additional level difference in conduction band on top of the one from the charge difference between La and Sr ions, which is considered here. Before presenting the explicit form of Hhop and Hon−site, let us discuss a difficulty one encounters when analyzing the hamiltonian Htot, in which the detail of Hhop and Hon−site does not matter. As can be seen from the geometry of the heterostructures we concern, our theoretical model is essentially equivalent to a two- dimensional quantum well. Therefore, there are two types of solutions to the one-electron equations: bound states, which decay as |z| →∞, and continuum states, which do not. The bound states give rise to sub- bands, some of which are partially occupied. The ground state is obtained by filling the lowest sub-bands up to the appropriate chemical potential (determined by charge neutrality); the interaction-related terms such as the electron self-energy are then recomputed and the procedure is repeated until self-consistency is obtained. Charge neutrality requires that the total density of electrons in the bound-state sub-bands equals the total density of La ions. However, the interplay between electron-La attraction and electron-electron repulsion leads (in almost all of the cases we have studied) to a very weak binding of the highest-lying electron states; indeed for large U the Fermi level of the partially filled sub-bands is only infinitesimally below the bottom of the continuum bands. Therefore, a large number of iterations is required to obtain accurate self consistency. In the HF analysis for the three-band model (basically T = 0), it is required to iterate 100–1000 times (the largest number of iterations is necessary for the phases with complicated spin and orbital orderings). In the DMFT analysis for the single- band model, 50–500 iterations are required. Again, the largest number of iterations is required for the magnetic phase at low temperature; fewer (20–50) are needed for the non-magnetic one at high temperature. Supercell techniques may alleviate this problem, at the cost a less precise treatment of the charge tails. In the following subsections, we present Hhop and Hon−site for", + "number of iterations is required for the magnetic phase at low temperature; fewer (20–50) are needed for the non-magnetic one at high temperature. Supercell techniques may alleviate this problem, at the cost a less precise treatment of the charge tails. In the following subsections, we present Hhop and Hon−site for the three-band and the single-band models. We also present the numerical methods employed to analyze these models. 2.1. Three-band model The relevant electronic orbitals are derived from the Ti t2g-symmetry d-states, and may be labeled as dxy, dxz, dyz. The derived bands41 for bulk materials are to a good approximation given by a nearest-neighbor tight binding model with hopping parameter of magnitude t ≃0.3 eV and spatial structure given by the Slater-Koster for- mula,42 so the dxy states disperse only in the xy plane etc. We take the form of the on-site interactions determined by Mizokawa et al.43 and adopt values as discussed below. Thus, Hhop = P α H(α) hop with H(xy) hop = −2t X ⃗kσ (cos kx + cos ky)d† xy σ⃗kdxy σ⃗k (3) and similarly for xz, yz. The onsite Hon−site is Hon−site = X i \u001a U X α niα↑niα↓+ (U ′ −J) X α>β,σ niασniβσ + U ′ X α̸=β niα↑niβ↓+ J X α̸=β d† iα↑diβ↑d† iβ↓diα↓ \u001b . (4) We have omitted a pair-transfer (J′) interaction which does not affect our results. For definiteness we follow other studies which employ the ratios U ′ = 7U/9 and J = U/9 which are very close to those determined by Mizokawa.43 Many workers have used the value U ∼5–6 eV ∼18t–20t estimated from high energy spectroscopies.32 However, optical conductivity studies of LaTiO3 and related compounds such as YTiO3 find rather small gaps, in the range 0.2–1 eV,44 suggesting U ∼2.5 eV ∼8t. In view of this uncertainty we investigate a range U from ∼6t–20t. One crucial aspect of the parameters chosen in Eq. (4) requires discussion: we recently found that, although in the isolated atom interactions always lead to orbital disproportionation (Hund’s second rule), in the solid environment this may or may not be the case according to the value of J/U.10 Let us consider minimizing the interaction energy, Eq. (4), with respect to a density matrix corresponding to a mean charge density per orbital nα and spin density per orbital mα. We find [n(m)tot = P α n(m)α] ⟨Hon−site⟩[nα, mα] = U \u0000n2 tot −m2 tot \u0001 4 + U −5J 2 X α>β nαnβ + U −J 2 X α>β mαmβ. (5) At fixed total density ntot and for a given number of orbitals norb, the term P α>β nαnβ is maximized by the uniform density nα = ntot/norb. Thus in a paramagnetic state (mα = 0 for all α) for J < U/5 an orbitally disproportionated state minimizes the interaction energy, whereas for J > U/5 a state of uniformly occupied orbitals minimizes the interaction energy. For spin polarized states the situation becomes more complicated, because the mα and nα are not independent (mα ≤nα). For fully spin polarized states (mα = nα), the", + "orbitally disproportionated state minimizes the interaction energy, whereas for J > U/5 a state of uniformly occupied orbitals minimizes the interaction energy. For spin polarized states the situation becomes more complicated, because the mα and nα are not independent (mα ≤nα). For fully spin polarized states (mα = nα), the condition for disproportionation becomes J < U/3. Therefore, our choice of parameter J = U/9 indicates that the orbitally disproportionated state becomes stable at large U, but a larger J would lead to a state with equal orbital occupancy. To study the properties of Htot for the three-band model, we employ the HF approximation replacing terms such as niασniβσ by niασ⟨niβσ⟩+ ⟨niασ⟩niβσ; orbitally off-diagonal expectation values ⟨d† iασdiβσ⟩of the type considered by Mizokawa43 and Mochizuki45 are stable only in the presence of a GdFeO3 type distortion which we do not consider. 2.2. Single-band model In some of our calculations, we consider a simpler model, which is the single-band model. The hamiltonian is a simplified representation of the systems studied in Ref. 14 with the orbital degeneracy neglected. We consider nearest-neighbor hopping as in the three-band model. Thus, Hhop and Hon−site are Hhop = −t X ⟨i,j⟩σ (d† iσdjσ + H.c.) (6) and Hon−site = U X i ni↑ni↓. (7) The single-band model can be studied by beyond-HF techniques. Here we use the dynamical-mean-field method in which the basic object of study is the electron Green’s function. In general, this is given by Gσ(⃗r, ⃗r′; ω) = [ω + µ −Hhop −Hpot −Σσ(⃗r, ⃗r′; ω)]−1, (8) with the chemical potential µ and the electron self-energy Σσ. Hpot represents the electrostatic potential from charge +1 counterions placed at La sites. In [001] heterostructures with either in-plane translational invariance or Ns-sublattice antiferromagnetism, the Green’s function and the self-energy are functions of the variables (z, η, z′, η′,⃗k∥) where η and η′(= 1, . . . , Ns) label the sublattice in layers z and z′, respectively, and ⃗k∥is a momentum in the (reduced) Brillouin zone. We approximate the self-energy as the sum of a static Hartree term ΣH σ (⃗r,⃗r) arising from the long-ranged part of the Coulomb interaction and a dynamical part ΣD σ (⃗r, ⃗r′; ω) arising from local fluctuations and which we assume that the self-energy is only dependent on layer z and sublattice η.6, 46, 47 Thus, the dynamical part of the self-energy is written as ΣD σ ⇒ΣD σ (z, η; ω). (9) The zη-dependent self-energy is determined from the solution of a quantum impurity model35 with the mean-field function fixed by the self-consistency condition Gimp σ (z, η; ω) = Ns Z d2k∥ (2π)2 Gσ(z, η, z, η,⃗k∥; ω). (10) 2.3. Computational Complexity and Impurity Solvers for DMFT In general for the heterostructure with L layers with Ns sublattices, one must solve L×Ns independent impurity models. Due to the self-consistency condition [cf. Eq. (10)] and to compute the charge density nσ(z, η) = − R dω π f(ω)ImGimp σ (z, η; ω) with f the Fermi distribution function, it is required to invert the (L × Ns)2 Green’s function", + "one must solve L×Ns independent impurity models. Due to the self-consistency condition [cf. Eq. (10)] and to compute the charge density nσ(z, η) = − R dω π f(ω)ImGimp σ (z, η; ω) with f the Fermi distribution function, it is required to invert the (L × Ns)2 Green’s function matrix at each momenta and frequency. This time consuming numerics restricts the size of the unit cell. In our work so far we have considered the commensurate magnetic states with up to two sublattices, Ns = 1 and 2, on each layer and with the charge density independent of the sublattices, i.e., paramagnetic (PM), ferromagnetic (FM) states, antiferromagnetic (AF) state where antiferromagnetic planes with moment alternating from plane to plane, and layer-AF state where FM planes with moment alternating from plane to plane. Note that the AF state extrapolates to the bulk AF state with magnetic vector ⃗q = (π, π, π) at n →∞. By symmetry, the number of quantum impurity models one must solve is reduced to L. However, solution of the impurity models is a time consuming task, and an inexpensive solver is required. In this study, we have employed two impurity solvers: (i) two-site DMFT48 and (ii) semiclassical approxima- tion (SCA).49 The two-site method is a simplified version of the exact-diagonalization method approximating a quantum impurity model as a cluster comprised of two orbitals. One orbital represents the impurity site which has the same interaction as in the lattice model and the other non-correlated (bath) site represents the temporal charge fluctuation.48 This method reproduces remarkably accurately the scaling of the quasiparticle weight and lower Hubbard band near the Mott transition. We have used the two-site method to investigate the dynamical properties of heterostructures in the paramagnetic state at T = 0. On the other hand, a small number of bath orbitals, here two, is known to be insufficient to describe the thermodynamics correctly.49 To investigate the magnetic behavior at non-zero temperature, we apply semiclassical approximation which we have recently devel- oped.49 In this approximation, two continuous Hubbard-Stratonovich transformations are introduced, coupling to spin- and charge-fields. When evaluating the partition function, only spin-fields at zero-Matsubara frequency are kept and saddle-point approximation is applied for charge-fields at given values of spin-fields. This method is found to be reasonably accurate to compute magnetic transition temperature because, in most of the correlated electron systems, very slow spin-fluctuation becomes dominant near the magnetic transition. In contrast to the two-site DMFT, the SCA can not reproduce quasiparticle peak at ω = 0. This is due to the neglect of quantum fluctuation of Hubbard-Stratonovich fields in the SCA. However, the SCA reproduces the spectral function in paramagnetic phase at not-too-low temperature and in the strong coupling regime, and in the magnetically ordered phase. 3. HARTREE-FOCK STUDY OF THREE-BAND MODEL In this section, we present the theoretical results of the realistic three-band-model based on the HF approximation. Our main focus is on the appearance of various spin and orbital orderings different from the bulk ordering. 3.1. Phase Diagram Figure 2 shows the calculated spin and orbital", + "STUDY OF THREE-BAND MODEL In this section, we present the theoretical results of the realistic three-band-model based on the HF approximation. Our main focus is on the appearance of various spin and orbital orderings different from the bulk ordering. 3.1. Phase Diagram Figure 2 shows the calculated spin and orbital phase diagram as a function of interaction strength and layer number. For reasons of computational convenience in scanning a wide range of parameters, we considered mainly phases with translation invariance in the xy plane, however for n = ∞and n = 1 we also considered an xy-plane two-sublattice symmetry breaking. We found, in agreement with previous calculations,43, 45, 50, 51 that the fully staggered phase is favored at n = ∞, but xy-plane symmetry broken states could not be stabilized in the one layer case. We have not yet studied more general symmetry breakings for intermediate layer numbers, but the physical arguments presented below strongly suggests that these phases only occur for larger numbers of layers (n >∼6). 300 $ )0 2' )0 22 \u0013 \u0018 \u0014\u0013 \u0014\u0018 \u0013\u0011\u0013 \u0013\u0011\u0015 \u0013\u0011\u0017 \u0013\u0011\u0019 \u0013\u0011\u001b \u0014\u0011\u0013 \u0014\u0003\u0012\u0003Q 8 \u0012 W )0 *\u001022 Figure 2. Ground state phase diagram as a function of the on-site Coulomb interaction U and inverse of the La layer number n computed by Hartree-Fock approximation. PMM: paramagnetic metallic state, (A)FM-OD: orbitally disordered state, ferromagnetic at n = 1 and layer-antiferromagnetic at n > 1, FM-OO: ferromagnetic state with two-dimensional orbital order, FM-G-OO: ferromagnetic state with (π, π, π) orbital order. We take U ′ = 7U/9, J = U/9, and Ec = 0.8. The triangle is the critical U above which (π, π, π) orbital ordering occurs for n = ∞. The broken line shows the expected phase transition to the (π, π, π) antiferromagnetic orbital ordering at finite n. Let us start with n = ∞limit corresponding to the bulk. The comparison to the bulk ordering is subtle. In bulk, LaTiO3 exhibits a (π, π, π) type antiferromagnetic ordering. Theoretical calculations (apparently confirmed by very recent NMR experiments, and x-ray and neutron diffraction experiments)52, 53 suggest a four-sublattice structure which is very close to (0, 0, π)-type orbital ordering54 but differing slightly from the large U ground state studied here. Stabilizing the observed state apparently requires a lattice distortion not included in the model studied here. As U is increased from zero the n →∞limit of the model considered here has a phase transition which we believe to be of second order to an incommensurate antiferromagnetic state with a wave vector which is an extremal spanning vector of the Fermi surfaces of the bands arising from two of the orbitals (say xz, yz) and which turns out to be very close to (0, 0, π). [In fact, for reasons of numerical simplicity we studied (0, 0, π) ordering and found a very weakly first order transition.] Within the model this transition is followed by a strongly first order transition to one of a degenerate manifold of states characterized by ferromagnetic spin order and (π, π, π) orbital order (triangle", + "of numerical simplicity we studied (0, 0, π) ordering and found a very weakly first order transition.] Within the model this transition is followed by a strongly first order transition to one of a degenerate manifold of states characterized by ferromagnetic spin order and (π, π, π) orbital order (triangle at n = ∞, U/t ≈7.5 in Fig. 2). We believe the (π, π, π)-orbital spin-ferro state we have found is a reasonable surrogate for the actual Mochizuki-Imada state found in experiment. Now we turn to the finite n region. We observe four phases: a small U phase with no broken symmetry [paramagnetic metallic phase (PMM)], and intermediate U phase with in-plane translation-invariance spin order, but no orbital order (OD), and a large U phase with both spin and orbital order [ferromagnetic orbitally ordered phase (FM-OO)]. The lower U transition line varies smoothly with layer number and at n →∞limit it asymptotes to the (0, 0, π) spin ordering in the bulk as discussed above. The n = 1 intermediate U phase is ferromagnetic (FM) whereas for n > 1 the intermediate U phase is antiferromagnetic (AFM). The essential reason for those orderings with in-plane translation-invariance is that for small n the charge density is spread in the z direction, so no layer has a density near 1. The larger U transition is essentially independent of layer number for n > 1. Only for sufficiently large n, is this state preempted by the bulk FM state with (π, π, π) OO found at large U. The essential point is that, for n < 6, the solution in the large U limit consists of several partially filled subbands, which have effectively minimized their interaction energy but which gain considerable kinetic energy from motion in the xy plane. Breaking of xy translation symmetry would reduce this kinetic energy gain without much decreasing the already saturated interaction energy while z-direction kinetic energy is quenched by the confining potential. Therefore, although computational difficulties have prevented us from precisely locating the FM-OO to bulk phase boundary we expect the dashed line in Fig. 2 is a reasonable estimate. For completeness, we have also shown the n →∞limit of the FM-OD to FM-OO phase boundary, calculated by suppressing the bulk phase. 3.2. Density Distribution and Metallic Edge We now turn to the spatial distribution of the charge density its relation to the metallic behavior. These are less sensitive to details of the ordered pattern and to the precise values of parameters. Typical results for density distribution are presented in Figs. 3 and 4 for the n = 6 heterostructure with U/t = 10. Charge +1 counterions are placed at z = ±0.5, ±1.5, ±2.5, thus, the regions at |z| < (>)3 correspond to LaTiO3 (SrTiO3). Figure 3 shows charge density distributions in the paramagnetic phase for different values of Ec = e2/(εat). Parameter values Ec = 2 and 0.25 correspond to the dielectric constant ε ∼6 and ∼48, respectively, with t ∼0.3 eV and a ∼3.9 ˚A. It is seen that the density distribution does not depend sensitively", + "charge density distributions in the paramagnetic phase for different values of Ec = e2/(εat). Parameter values Ec = 2 and 0.25 correspond to the dielectric constant ε ∼6 and ∼48, respectively, with t ∼0.3 eV and a ∼3.9 ˚A. It is seen that the density distribution does not depend sensitively on Ec, thus, neither does the HF result for the phase diagram. However, it is expected that Ec affects the boundary between phases with and without in-plane symmetry because the stability of the in-plane symmetry broken phase is sensitive to how the density is close to 1 which is required for the bulk like ordering. This issue is discussed in detail for the single-band model in Ref. 37. Figure 4 presents density profile (open squares) of the orbitally ordered ferromagnetic state with Ec = 0.8. It is seen that difference from the paramagnetic state shown in Fig. 3 is small. At the interface between the two regions |z| = 3, we observe about three-unit-cell-wide crossover regime where the density drops from ∼1 to ∼0. This crossover region turned out to support metallic behavior, thus we term it a metallic edge. Within the HF method we use here, physics of the metallic edge is manifested as follows. There are many bound-state solutions (wave functions decaying as |z| is increased away form the heterostructure), whose dis- persion in the in-plane direction gives rise to subbands. For thin heterostructures, all subbands are partially filled (implying metallic behavior in the heterostructure plane) whereas for thick heterostructures (in ordered phases), some subbands are fully filled and some are partly filled. The fully filled subbands have z-direction wave functions implying charge density concentrated in the middle of the heterostructure, whereas the partially filled bands have charge concentrated near the edges. The occupancy of partially filled bands is computed and presented as open circles in Fig. 4. We observe partially filled bands corresponding to metallic behavior at the crossover regime. Summarizing this section, we applied the HF approximation to model heterostructures comprised of three- band Hubbard model. The ground-state phase diagram shows that thin heterostructures exhibit spin and orbital \u0010\u001b \u0010\u0019 \u0010\u0017 \u0010\u0015 \u0013 \u0015 \u0017 \u0019 \u001b \u0013\u0011\u0013 \u0013\u0011\u0015 \u0013\u0011\u0017 \u0013\u0011\u0019 \u0013\u0011\u001b \u0014\u0011\u0013 (F \u0015 \u0014 \u0013\u0011\u001b \u0013\u0011\u0018 \u0013\u0011\u0015\u0018 QWRW ] Figure 3. Total density profile of n = 6 heterostructure with different values of Ec indicated. U/t = 10, and a para- magnetic state is assumed. Ec = 2 and 0.25 correspond to the dielectric constant ε ∼6 and ∼48, respectively, with t ∼0.3 eV and a ∼3.9 ˚A. \u0010\u001b \u0010\u0019 \u0010\u0017 \u0010\u0015 \u0013 \u0015 \u0017 \u0019 \u001b \u0013\u0011\u0013 \u0013\u0011\u0015 \u0013\u0011\u0017 \u0013\u0011\u0019 \u0013\u0011\u001b \u0014\u0011\u0013 QSDUW QWRW QWRW\u000f\u0003QSDUW ] Figure 4. Total density profile (open squares) and den- sity in partially occupied bands npart (open circles) of het- erostructure with n = 6, U/t = 10 and Ec = 0.8 calculated in orbitally ordered ferromagnetic phase. Charge +1 coun- terions are placed at z = ±0.5, ±1.5, ±2.5, so the electronic (B) sites are at the integer values of z. orderings different from the bulk ordering. The charge", + "erostructure with n = 6, U/t = 10 and Ec = 0.8 calculated in orbitally ordered ferromagnetic phase. Charge +1 coun- terions are placed at z = ±0.5, ±1.5, ±2.5, so the electronic (B) sites are at the integer values of z. orderings different from the bulk ordering. The charge density at the edge region where the density drops from ∼1 to ∼0 is dominated by the partially filled bands indicating metallic behavior in this region, although the two component compounds are insulating in bulk. These results demonstrate the concept of electronic reconstruction in correlated-electron heterostructures. As will be discussed in the next section, details of phase boundaries and ordering patterns will be changed when beyond-HF methods are applied. Furthermore, values of on-site interaction in real materials may be changed by changing the thickness and changing the screening effect. However, the general conclusion that non-bulk phases occur and that an interesting series of phase transitions may take place as the layer thickness is varied are robust. 4. DYNAMICAL-MEAN-FIELD STUDY OF SINGLE-BAND MODEL This section presents DMFT studies of single-band heterostructures. First, we present results for the spatial variation of dynamical properties of electrons such as single-particle spectral functions. It is shown that the quasiparticle appearing at interfaces between Mott-insulator region and band-insulator region is only moderately correlated and gives rise to metallic behavior. Second, we discuss magnetic orderings in the heterostructures which are different from the bulk ordering. Theoretical results confirm the important results obtained by the HF approximation shown in the previous section, i.e., different phases in thin heterostructures than in the bulk, and metallic edge, but show the significant improvement providing the reasonable estimates of the transition temperature and new insights into the spatial variation of the order parameter. 4.1. Heavy Quasiparticle and Metallic Edge Here, we apply the two-site DMFT to the single-band heterostructures to investigate the dynamical properties of heterostructures. We consider paramagnetic states at T = 0. This is because we are interested in the crossover behavior between a Mott-insulating state, where the insulating behavior purely comes from the electron correlation, and the HF method can not be applied, to the others. One of the most useful observable to see the dynamical property of the correlated electron is the single particle spectral function. The spectral functions are in principle measurable in photoemission or scanning tunneling microscopy. Numerical results for the layer-resolved spectral function A(z; ω) = −1 π R d2k∥ (2π)2 ImG(z, z,⃗k∥; ω +i0+) for a 10-layer heterostructure with different values of U are presented in Fig. 5. The dimensionless parameter for the long-ranged Coulomb interaction is Ec = 0.8. The left panel shows results for the weak coupling (U/t = 10), and the right panel for the strong coupling (U/t = 16 about 10% greater than the critical value which drives the Mott transition in a bulk system described by H with n = ∞). The critical value for the bulk Mott transition is estimated to be Uc/t ≈14.7 by the two-site DMFT. Outside the heterostructure (|z| ≫6), the spectral function is essentially identical in", + "than the critical value which drives the Mott transition in a bulk system described by H with n = ∞). The critical value for the bulk Mott transition is estimated to be Uc/t ≈14.7 by the two-site DMFT. Outside the heterostructure (|z| ≫6), the spectral function is essentially identical in form to that of the free tight-binding model Hband for both the weak coupling and strong coupling results. The electron density is negligible, as can be seen from the fact that almost all of the spectral function lies above the chemical potential. As one approaches the heterostructure (|z| = 6), the spectral function begins to broaden. For the weak coupling case, spectral functions at |z| < 6 are also quite similar to the results of the HF analysis except for tiny peaks outside of central quasiparticle band corresponding to the upper- and lower-Hubbard bands. On the other hand for the strong coupling case, spectral weight around ω = 0 begins to decrease rapidly and the characteristic strong correlation structure of lower and upper Hubbard bands with a central quasiparticle peak begins to form. The sharp separation between these features is an artifact of the two-site DMFT [as is, we suspect, the shift in energy of the upper (empty state) Hubbard band for z = 4, 5]. Experience with bulk calculations suggests that the existence of three features and the weight in the quasiparticle region are reliable. Towards the center of the heterostructure, the weight in the quasiparticle band becomes very small, indicating nearly insulating behavior. For very thick heterostructures, we find the weight approaches 0 exponentially. The behavior shown in Fig. 5 is driven by the variation in density caused by leakage of electrons out of the heterostructure region. Figure 6 shows as open squares the numerical results for the charge-density distribution ntot(z) for the heterostructure whose photoemission spectra are shown in Fig. 5. One sees that in the center of the heterostructure (z = 0) the charge density is approximately 1 per site, and that there exists an edge region, \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0010\u0014\u0013 \u0013 \u0014\u0013 \u0015\u0013 \u0013\u0011\u0013 \u0013\u0011\u0014 ] \u001a ] \u0019 ] \u0018 ] \u0017 ] \u0016 ] \u0015 ] \u0014 ] \u0013 ω\u0012W $ ] ω \u0003 \u0003 \u0003 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 ] \u0014\u0013 ] ] \u001b w \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0010\u0014\u0013 \u0013 \u0014\u0013 \u0015\u0013 \u0013\u0011\u0013 \u0013\u0011\u0014 ] \u001a ] \u0019 ] \u0018 ] \u0017 ] \u0016 ] \u0015 ] \u0014 ] \u0013 ω\u0012W \u0003 \u0003 $ ] ω \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 ] \u0014\u0013 ] ] \u001b w Figure 5. Layer-resolved spectral function calculated for 10-layer heterostructure with Ec = 0.8. Left panel: U/t = 10, right panel: U/t = 16. Charge +1 counterions are placed at z = ±0.5, . . . , ±4.5 so the electronic (B) sites are at integer values of z. \u0010\u0014\u0013 \u0010\u0018 \u0013 \u0018 \u0014\u0013 \u0013\u0011\u0013 \u0013\u0011\u0015 \u0013\u0011\u0017 \u0013\u0011\u0019 \u0013\u0011\u001b \u0014\u0011\u0013 QWRW QFRK QWRW\u000f\u0003QFRK", + "0.8. Left panel: U/t = 10, right panel: U/t = 16. Charge +1 counterions are placed at z = ±0.5, . . . , ±4.5 so the electronic (B) sites are at integer values of z. \u0010\u0014\u0013 \u0010\u0018 \u0013 \u0018 \u0014\u0013 \u0013\u0011\u0013 \u0013\u0011\u0015 \u0013\u0011\u0017 \u0013\u0011\u0019 \u0013\u0011\u001b \u0014\u0011\u0013 QWRW QFRK QWRW\u000f\u0003QFRK ] Figure 6. Total charge density (open squares) and charge density from the coherent part near the Fermi level (open circles). For comparison, the total charge density calculated by applying the HF approximation to the Hamiltonian is shown as filled symbols. The parameters are the same as in the right panel of Fig. 5. of about three-unit-cell width, over which the density drops from ∼1 to ∼0. The overall charge profile is determined mainly by the self-consistent screening of the Coulomb fields which define the heterostructure, and is only very weakly affected by the details of the strong on-site correlations (although the fact that the correlations constrain ntot < 1 is obviously important). To show this, we have used the HF approximation to recalculate the charge profile: the results are shown as filled circles in Fig. 6 and are seen to be almost identical to the DMFT results. In order to study the metallic behavior associated with the quasiparticle subband, we computed the charge density from the quasiparticle bands ncoh by integrating A(z; ω) from ω = 0 down to the first point at which A(z; ω) = 0. Results are shown as open circles in Fig. 6. It is obvious that these near-Fermi-surface states contain a small but non-negligible fraction of the total density, suggesting that edges should display relatively robust metallic behavior. The results represent a significant correction to the HF calculation shown in Fig. 4, which leads, in the edge region, to a metallic quasiparticle density essentially equal to the total density. This is because the HF approximation does not give mass renormalization. Calculations, not shown here, of the dispersion of the quasiparticle subbands find that the mass renormalization is, to a good approximation, m∗/m ∼ntot/ncoh. 4.2. Magnetic Ordering at Finite Temperature Now, we turn to the magnetic ordering of the single-band heterostructures at non-zero temperature. The HF approximation provides a very poor approximation to the behavior in this region. We also discuss differences between DMFT and HF. Figure 7 shows our calculated phase diagram in the interaction-temperature plane for heterostructures with various thicknesses. The one-layer heterostructure is PM at weak to moderate interactions, and FM at strong interactions. The two- and three-layer heterostructures are AF at weak to intermediate interaction, and become FM at stronger interactions with almost the same TC for n = 2 and 3. The phase diagram displays regions where both TC and TN > 0 (FM and AF both locally stable); in these regions the phase with the higher transition temperature has the lower free energy and is the one which actually occurs. HF studies of the single-orbital model find a layer-AF phase. This phase is not found in our DMFT analysis. Note that an antiferromagnetic ordering in a two-dimensional system occurs only at", + "the phase with the higher transition temperature has the lower free energy and is the one which actually occurs. HF studies of the single-orbital model find a layer-AF phase. This phase is not found in our DMFT analysis. Note that an antiferromagnetic ordering in a two-dimensional system occurs only at T = 0. The non-zero TN is an artifact of the mean-field nature of the DMFT. In a real system, slow antiferromagnetic fluctuations are expected to develop for temperatures below the TN determined by the DMFT. As will be shown below, the ferromagnetism is an interface effect. We expect that at large U, very thick heterostructures will be AF in the center, but with a FM surface layer. Antiferromagnetic N´eel temperature TN is found to be strongly dependent on the layer thickness; it increases with the increase of layer thickness. Note that TN’s are substantially reduced from the bulk value, TN ∼6t2/U at strong-coupling regime (see a light solid line in Fig. 7), due to the smaller charge density per site. Further, the in-plane symmetry broken phase is found to become unstable by increasing the dielectric constant and broadening the charge density profile. (For details, see Ref. 37.) We now turn to the spatial variation of the magnetization density. As examples, numerical results for a 4-layer heterostructure (counterions at z = ±0.5 and ±1.5) with Ec = 0.8 are presented in Fig. 8. The upper panel of \u0013 \u0018 \u0014\u0013 \u0014\u0018 \u0015\u0013 \u0015\u0018 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0015 \u0013\u0011\u0016 \u0013\u0011\u0017 \u0013\u0011\u0018 )0 $) … Q 7& 71 \u0014 \u0015 \u0016 7\u0012W 8\u0012W Figure 7. Magnetic transition temperatures of heterostructures with various thicknesses n as functions of interaction strength. Ec = 0.8. Filled symbols: Ferromagnetic Curie temperature TC, open symbols: Antiferromagnetic N´eel temperature TN. Note that the n = 1 heterostructure does not exhibit a N´eel phase. Note that, where both phases are locally stable, the phase with higher TC(TN) has the lower free energy. For comparison, TN for bulk AF ordering computed by the same method is shown as the light solid line. \u0013\u0011\u0013 \u0013\u0011\u0015 \u0013\u0011\u0017 \u0013\u0011\u0019 \u0013\u0011\u001b \u0014\u0011\u0013 '0)7 +) 8\u0012W \u0015\u0017 7\u0012W \u0013\u0011\u0014 P \u0010\u0019 \u0010\u0017 \u0010\u0015 \u0013 \u0015 \u0017 \u0019 \u0010\u0013\u0011\u0018 \u0013\u0011\u0013 \u0013\u0011\u0018 \u0014\u0011\u0013 8\u0012W \u0019 7\u0012W \u0013\u0011\u0013\u0018 PVWDJJ QWRW PVWDJJ\u0003\u000f\u0003QWRW ] )0 $) Figure 8. Magnetization density of 4-layer heterostruc- ture. Counterions are placed at z = ±0.5, ±1.5. Ec = 0.8. Upper panel: Magnetization m in a FM state for U/t = 24 and T/t = 0.1. Lower panel: In-plane staggered magneti- zation mstagg in an AF state for U/t = 6 and T/t = 0.05. Filled (open) circles are the results by the DMFT (HF). For comparison, charge density ntot computed by the DMFT is also shown in the lower panel (filled squares). Note that the staggered magnetization in the outer layers (|z| ≥2), and the outermost layers (|z| = 1) have the same sign. \u0013\u0011\u0013\u0013 \u0013\u0011\u0013\u001a \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0015 \u0010\u0014\u0013 \u0013 \u0014\u0013 \u0015\u0013 \u0013\u0011\u0013\u0013 \u0013\u0011\u0013\u001a ] \u0013 ] \u0014 $σ ] ω ] \u0015 8\u0012W \u0015\u0017 7\u0012W \u0013\u0011\u0014 ↑ ↓ ] \u0016 \u0013\u0011\u0013\u0013 \u0013\u0011\u0015\u0018 \u0013\u0011\u0013\u0013", + "in the outer layers (|z| ≥2), and the outermost layers (|z| = 1) have the same sign. \u0013\u0011\u0013\u0013 \u0013\u0011\u0013\u001a \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0013 \u0013\u0011\u0014 \u0013\u0011\u0015 \u0010\u0014\u0013 \u0013 \u0014\u0013 \u0015\u0013 \u0013\u0011\u0013\u0013 \u0013\u0011\u0013\u001a ] \u0013 ] \u0014 $σ ] ω ] \u0015 8\u0012W \u0015\u0017 7\u0012W \u0013\u0011\u0014 ↑ ↓ ] \u0016 \u0013\u0011\u0013\u0013 \u0013\u0011\u0015\u0018 \u0013\u0011\u0013\u0013 \u0013\u0011\u0014\u0018 \u0013\u0011\u0013\u0013 \u0013\u0011\u0014\u0018 \u0013\u0011\u0016\u0013 \u0010\u0018 \u0013 \u0018 \u0014\u0013 \u0013\u0011\u0013\u0013 \u0013\u0011\u0015\u0018 ω \u0012W ] \u0016 ] \u0015 ] \u0014 ] \u0013 $σ ]\u000f\u0014 ω w 8\u0012W \u0019 7\u0012W \u0013\u0011\u0013\u0018 w )0 $) Figure 9. Layer- and sublattice-resolved spectral func- tions as functions of real frequency ω for 4-layer het- erostructure. Ec = 0.8. Upper panel: Ferromagnetic state at U/t = 24 and T/t = 0.1. Lower panel: Antiferromag- netic state at U/t = 6 and T/t = 0.05. Sublattice η = 1. Solid (broken) lines are for up (down) spin electrons. For sublattice η = 2 in AF state, up and down electrons are interchanged. Fig. 8 shows the magnetization in the FM state. In the DMFT (filled circles), only the layers near the interfaces (|z| ∼2) have large polarization and inner layers in the heterostructure have small moments. This explains the weak n-dependence of TC of thick heterostructures (see the upper panel of Fig. 7). In the HF (open circles), all layers in the heterostructure are highly polarized. In contrast, in AF states, the in-plane staggered magnetization computed by the DMFT and the HF agree well as shown in the lower panel of Fig. 8. For comparison, the total charge density is also plotted (filled squares). The in-plane staggered magnetization is large only at inner layers where the charge density is close to 1. Note that the staggered magnetization in the outer layers (|z| ≥2) has the same sign as in the layers at |z| = 1 indicating that the outer layers are not intrinsically magnetic. The spin distributions presented in Fig. 8 can be understood from the single-particle spectral functions. In Fig. 9 are presented the DMFT results for the layer- and sublattice-resolved spectral functions Aσ(z, η; ω) = −1 πImGimp σ (z, η; ω+i0+) for the FM (upper panel) and the AF (lower panel) states of 4-layer heterostructure with the same parameters as in Fig. 8. These quantities can in principle be measured by spin-dependent photoemission or scanning tunneling microscopy. As noticed in Sec. 4.1, spectral function outside of the heterostructure (|z| ≫2) is essentially identical to that of the free tight-binding model Hband, and electron density is negligibly small. Approaching the interfaces (|z| = 2), the spectral function shifts downwards and begins to broaden. In the FM case, magnetic ordering is possible only near the interface (|z| ∼2) where the charge density is intermediate. Inside the heterostructure (|z| < 2), clear Hubbard gap exists due to the large U and uniform polarization is hard to achieve. On the contrary, high charge density is necessary to keep the staggered magnetization in the AF case as seen as a difference between up and down spectra in the lower panel of Fig. 9. Summarizing this section, by applying the DMFT to the", + "and uniform polarization is hard to achieve. On the contrary, high charge density is necessary to keep the staggered magnetization in the AF case as seen as a difference between up and down spectra in the lower panel of Fig. 9. Summarizing this section, by applying the DMFT to the single-band heterostructure, we confirmed two important results obtained in the previous section on HF analysis: thin heterostructures show different magnetic orderings and an approximately three-unit-cell-wide interface region becomes metallic. Using the DMFT, we obtained new information about correlated-heterostructure behaviors: metallic edge involves quasiparticle bands with moderate mass enhancement, ferromagnetic Curie temperature is estimated to be low compared with the bulk N´eel temperature, ferromagnetic moment is concentrated in the interface region where there is intermediate charge density while antiferromagnetic staggered moment is concentrated in the region where the density is close to the one for the bulk Mott insulating compound. Thus, electronic reconstruction is a generic property of the correlated-electron heterostructures. 5. CONCLUSION This paper summarizes our theoretical studies of the electronic properties of a correlated heterostructure model involving n layers of a material which in bulk is a Mott insulator, embedded in an infinite band insulator. The specific features of the model we study were chosen to reproduce the LaTiO3/SrTiO3 heteostructure system studied by Ohtomo et al., but we hope that our results will shed light also on the more general question of the physics of interfaces between strongly correlated and weakly correlated systems. A crucial feature of the experimental LaTiO3/SrTiO3 system studied by Ohtomo et al. is the almost perfect lattice match between the two systems. These authors argued that this implies that the only difference between the Mott insulating and band insulating regions arises from the different charge of the La3+ and Sr2+; in particular the crystal structure and atomic positions are expected to remain relatively constant throughout the heterostructure. Of course, the asymmetry present at the LaTiO3/SrTiO3 interface must induce some changes in atomic positions: a TiO6 octahedron is negatively charged, and so if it sits between a Sr plane and a La plane it will be attracted to the latter, and also distorted, because the positively charged Ti will tend to move in the opposite direction. (When one uses ε = 1, the electrostatic force exerted by a LaO plane on the neighboring O2−ion is estimated to be ≈20 eV/˚A !) The experimentally determined Ti-Ti distances shown in Fig. 1 of Ref. 14, along with the distortion in that paper, suggests that the changes in Ti-Ti distance are negligible. In this circumstance, changes in O position along the Ti-O-Ti bond change hoppings only in second order. We therefore neglected these effects and assumed that the electronic hoppings and interaction parameters remain invariant across the heterostructure. However, we emphasize that properly accounting for the effect of atomic rearrangements inevitably present at surface and interface is crucial. We further note that lattice distortions appear to be important in stabilizing the observed bulk state, but may be pinned in a heterostructure. Extending our results to include these effects is an important", + "that properly accounting for the effect of atomic rearrangements inevitably present at surface and interface is crucial. We further note that lattice distortions appear to be important in stabilizing the observed bulk state, but may be pinned in a heterostructure. Extending our results to include these effects is an important open problem. In the calculations reported here, the heterostructure is defined only by the difference (+3 vs +2) of the La and Sr charge. The calculated electronic charge density is found to be controlled mainly by electrostatic effects (ionic potentials screened by the electronic charge distribution). Results, such as shown in Figs. 3 and 4 for the three-band model or Fig. 6 for the single-band model, are representative of results obtained for a wide range of on-site interaction U, long-ranged Coulomb interaction Ec, and different theoretical methods. We find generally that significant leakage of charge into the band insulator region occurs. The width of the transition region must depend on the relative strength of the z-direction hoppings and the confining potential. For the parameters studied here, the transition region where the charge density changes from the value for the bulk Mott-insulator ∼1 to the one for the band-insulator ∼0 is about three layers. The spreading of the electronic charge controls the electronic properties of the heterostructure. Using Hartree-Fock and dynamical-mean-field approximations, we demonstrated that spin (and orbital) order- ings in thin heterostructures differ from that in the bulk. This behavior originates from the lower charge density in the heterostructure due to the leakage of electrons from the Mott-insulating region to the band-insulating region. The dynamical properties of correlated-electron heterostructures were calculated using the dynamical- mean-field method. Our results show how the electronic behavior evolves from the weakly correlated to the strongly correlated regions, and in particular, confirms the existence of an approximately three-unit-cell-wide crossover region in which a system, insulating in bulk, can sustain metallic behavior. We found that even in the presence of very strong bulk correlations, the metallic edge behavior displays a moderate mass renormalization. In Ref. 36, we have also discussed how the magnitude of the renormalization is affected by the spatial structure of the quasiparticle wave function and determined how this renormalization affects physical observables such as the optical conductivity. Further, in Ref. 37, we have investigated how changes in the charge density distribution affects the magnetic transition temperature. Being based on the theoretical studies, we would like to propose a new and important concept in the correlated- electron interface problem: electronic reconstruction which means electronic phase behavior at interfaces differs from that in the bulk. Electronic reconstruction is originally proposed by Hesper et al. to describe the rear- rangement of the electronic charge in the presence of a polar surface of C60 films.8 We suggest this phrase be applied more generally to the electronic phase behavior. This contrasts with the ordinary lattice reconstruction implying the appearance of the different lattice structure at the interface than in the bulk of un-correlated or weakly correlated materials. Finally, important future directions for research include re-examination of the phase diagram of", + "be applied more generally to the electronic phase behavior. This contrasts with the ordinary lattice reconstruction implying the appearance of the different lattice structure at the interface than in the bulk of un-correlated or weakly correlated materials. Finally, important future directions for research include re-examination of the phase diagram of three-band model using beyond Hartree-Fock techniques, and generalization of the results presented here to more realistic cases. 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Braden, “Crystal and magnetic structure of LaTiO3: Evidence for nondegen- erate t2g orbitals,” Phys. Rev. B 68, 060401(R), 2003. 54. M. Mochizuki and M. Imada, “Orbital-spin structure and lattice coupling in RTiO3 where R = La, Pr, Nd, and Sm,” Phys. Rev. Lett. 91, 167203, 2003. 55. C. Lin, A. J. Millis, and S. Okamoto, “Dynamical mean field study of Manganite superlattices,” (in prepa- ration). 56. M. Takizawa, T. Wadati, M. Kobayashi, S. Tanaka, S. Yagi, S. Hashimoto, T. Yoshida, A. Fujimori, S. Chikamatsu, H. Kumigashira, S. Ojima, K. Shibuya, S. Mihara, T. Ohonishi, M. Lippmaa, M. Kawasaki, H. Koinuma (private communication).", + "arXiv:cond-mat/0208595v1 [cond-mat.str-el] 30 Aug 2002 Spectral Functions and Pseudogap in Models of Strongly Correlated Electrons∗ P. Prelovˇsek and A. Ramˇsak Faculty of Mathematics and Physics, University of Ljubljana, 1111 Ljubljana, Slovenia J. Stefan Institute, University of Ljubljana, 1111 Ljubljana, Slovenia The theoretical investigation of spectral functions and pseudogap in systems with strongly correlated electrons is discussed, with the emphasis on the single-band t-J model as relevant for superconducting cuprates. The evidence for the pseudogap features from numerical studies of the model is presented. One of the promising methods to study spectral functions is the method of equations of motion. The latter can deal systematically with the local constraints and projected fermion operators inherent for strongly correlated electrons. In the evaluation of the self energy the decoupling of spin and single-particle fluctuations is performed. In an undoped anti- ferromagnet the method reproduces the selfconsistent Born approximation (SCBA). For finite doping the approximation evolves into a paramagnon contribution which retains large incoherent contribution in the hole part. On the other hand, the contribution of longer-range spin fluctuations is essential for the emergence of the pseudogap. The latter shows up at low doping in the effective truncation of the large Fermi surface, reduced elec- tron density of states and at the same time reduced quasiparticle density of states at the Fermi level. PACS numbers: 71.27.+a, 72.15.-v, 71.10.Fd 1. Introduction One of the central questions in the theory of strongly correlated electrons is the nature of the ground state and of low energy excitations. Experiments in many novel materials with correlated electrons [1] reveal even in the ’nor- mal’ metallic state striking deviations from the usual Fermi-liquid univer- sality as given by the phenomenological Landau theory involving quasipar- ticles (QP) as the well defined fermionic excitations even in the presence ∗Presented at the Strongly Correlated Electron Systems Conference, Krak´ow 2002 (1) 2 Pseudogap printed on October 30, 2018 of Coulomb interactions. The focus has been and still remains on super- conducting cuprates where there exists now an abundant and consistent experimental evidence for very anomalous low-energy properties [1], besides the most evident open question of the origin of the high-Tc superconductiv- ity. In particular, the attention in the last decade has been increasingly de- voted to the underdoped cuprates, where experiments reveal characteristic ’pseudogap’ temperatures, which show up crossovers where particular prop- erties change quantitatively. As schematically presented in Fig. 1, there seems to be an indication for two crossover scales T ∗[2] and Tsg [1]. The existence of both is still widely debated, in particular whether both could be a manifestation of the same underlying mechanism. Nevertheless we refer to them (as usually in the literature) as the (larger) pseudogap scale T ∗and the spin-gap scale Tsg for the lower one. Fig. 1. Schematic electronic phase diagram of cuprates The T ∗scale [2] shows up clearly as the maximum of the spin suscep- tibility χ0(T = T ∗) [3]. The in-plane resistivity ρ(T) is linear ρ ∝T for T > T ∗and decreases more steeply for T < T ∗. The same T ∗appears in", + "phase diagram of cuprates The T ∗scale [2] shows up clearly as the maximum of the spin suscep- tibility χ0(T = T ∗) [3]. The in-plane resistivity ρ(T) is linear ρ ∝T for T > T ∗and decreases more steeply for T < T ∗. The same T ∗appears in the anomalous T-dependence of the Hall constant RH(T) for T < T ∗ [1]. For the theoretical considerations the most straightforward signature of pseudogap is the reduction of the specific heat coefficient γ = CV /T be- low T ∼T ∗in the underdoped cuprates [4]. Namely, within normal Fermi liquids γ directly measures the quasiparticle (QP) density of states. The spin-gap crossover Tsg has been identified in connection with the decrease of the NMR relaxation rate 1/T1 for T < Tsg [1], related to the reduction of low-energy spin excitations. Even more striking is the observation of Pseudogap printed on October 30, 2018 3 the leading-edge shift [5] in the angle-resolved photoemission spectroscopy (ARPES) measurements at T > Tc, a feature interpreted as a d-wave SC gap persisting within the normal phase. It seems to some extent plausible that the T ∗crossover is related to the onset of short-range antiferromagnetic (AFM) correlations for T < T ∗, since χ(T) in an undoped AFM has a maximum at T ∗∼2J/3 the temperature corresponding to a gradual transition from a disordered paramagnet to the one with short-range AFM correlations. In this contribution we concentrate on our theoretical results which confirm and explain the existence of the pseudogap in the model relevant for cuprates, i.e. the t-J model on a 2D square lattice. It should be however pointed out that there there are also several alternative theoretical proposals, which do not directly invoke AFM spin correlations. The single-particle spectral function A(k, ω) and its properties are of crucial importance, since their full knowledge would essentially clarify most open questions concerning the anomalous character of strongly correlated electron systems. In recent years there has been an impressive progress in ARPES experiments [5, 6, 1] (in particular for cuprate materials) which in principle yield a direct information on A(k, ω). In most investigated Bi2Sr2CaCu2O2+δ (BSCCO) [5] ARPES shows quite a well defined large Fermi surface (FS) in the overdoped and optimally doped samples at T > Tc, whereby the low-energy behavior with increasing doping in the overdoped regime qualitatively approaches (but does not in fact reach) that of the nor- mal Fermi-liquid with underdamped quasiparticle (QP) excitations. On the other hand, in the underdoped BSCCO QP dispersing through the Fermi surface (FS) are resolved by ARPES only in parts of the large FS, in par- ticular along the nodal (0, 0)-(π, π) direction [7, 5], indicating that the rest of the large FS is truncated [8], i.e. either fully or effectively gaped. At the same time near the (π, 0) momentum ARPES reveals a hump at ∼100 meV [7, 5], which is consistent with large pseudogap scale T ∗. Spectral proper- ties for La2−xSrxCu2O4 (LSCO), as revealed by ARPES [6] appear to some extent different from BSCCO,", + "either fully or effectively gaped. At the same time near the (π, 0) momentum ARPES reveals a hump at ∼100 meV [7, 5], which is consistent with large pseudogap scale T ∗. Spectral proper- ties for La2−xSrxCu2O4 (LSCO), as revealed by ARPES [6] appear to some extent different from BSCCO, presumably due to the crucial role of stripe structures in the LSCO in the regime of intermediate doping. Still they again reveal a truncated FS at low-doping and even the existence of QP along the nodal direction. The prototype single-band model relevant for cuprates which takes ex- plicitly into account strong correlations is the t-J model, derived originally by Chao, Spa lek and Ole´s [9] H = − X i,j,s tij˜c† js˜cis + J X ⟨ij⟩ (Si · Sj −1 4ninj), (1) where fermionic operators are projected ones not allowing for the double 4 Pseudogap printed on October 30, 2018 occupancy of sites, i.e., ˜c† is = (1 −ni,−s)c† is. (2) Longer range hopping appears to be important for the proper description of spectral function in cuprates, in particular it is invoked to explain the dif- ference between electron-doped and hole-doped materials both in the shape of the FS at optimum doping materials [10] as well as for the explanation of the ARPES of undoped insulators [11, 10], we consider besides tij = t for the n.n. hopping also tij = t′ for the n.n.n. hopping on a square lattice. Note that t′ < 0 for the hole-doped cuprates. There have been so far numerous theoretical studies of the t-J model on square lattice and related Hubbard model at large Coulomb repulsion U ≫t , as relevant to cuprates, using both analytical approaches as well as numerical techniques for finite size systems. Still analytical approxima- tions to spectral properties have proved to be very delicate, in particular with respect to the question of the emerging pseudogap at lower doping. So there are much fewer studies which give some answers on latter questions within microscopic models close to cuprates. The importance of AFM spin correlations for the emergence of the (large) pseudogap is found in the nu- merical studies [12, 13, 14] and in phenomenological model studies [15]. The renormalization group studies of the Hubbard model [16] also reveal the in- stability of the normal Fermi liquid close to the half-filled band (insulator) and a possible truncation of the Fermi surface. In the following we describe some evidence for the pseudogap within the t-J model obtained via finite-size studies and a novel approach to spectral functions using the method of equations of motion. 2. Evidence for the pseudogap from numerical studies We introduced some years ago a numerical method, i.e. the finite- temperature Lanczos method (FTLM) [17, 14], which is particularly useful for studying finite-size model systems of correlated electrons at T > 0. The technical advantage of the method is that it is comparable in efficiency to ground state calculations. T > 0 results for static and dynamical quantities are of interest in themselves, allowing to follow the T-variation of properties whereby", + "studying finite-size model systems of correlated electrons at T > 0. The technical advantage of the method is that it is comparable in efficiency to ground state calculations. T > 0 results for static and dynamical quantities are of interest in themselves, allowing to follow the T-variation of properties whereby some of them are meaningful only at T > 0, e.g. the entropy, the specific heat, the d.c.resistivity etc. On the other hand, the usage of finite but small T > 0 represents a proper approach to more reliable ground state calculations in small systems. The most straightforward evidence for a pseudogap within the planar t-J model allowing the comparison with experiments appears in the uniform static spin-suceptibility χ0(T) [18, 14]. Results for various hole concentra- tions ch = Nh/N in a system with N = 20 sites are presented in Fig. 2. Pseudogap printed on October 30, 2018 5 0.0 0.2 0.4 0.6 0.8 1.0 T/t 0.0 0.2 0.4 0.6 χ0t ch=0.25 ch=0 J/t Fig. 2. Uniform susceptibility χ0(T ) at several hole doping ch. Results are for J/t = 0.3. It is evident that the maximum T ∗being related to the AFM exchange T ∗∼2J/3 in an undoped AFM gradually shifts down with doping and finally disappears at ’critical’ ch = c∗ h ∼0.15. Obtained results are qualita- tively as well as quantitatively consistent with experiments in cuprates, e.g. LSCO [3]. Another quantity relevant for comparison with the analytical approach further on is the single-particle density of states (DOS) N(ω) [13, 14]. We present in Fig. 3 numerical results for DOS [19] as a function of T for systems with N = 18, 20 sites and two lowest nonzero meaningful hole concentrations ch ∼0.05, 0.11. At smallest ch = 0.05 there is a pronounced pseudogap at ω ∼0 which closes with increasing T ∼T ∗∼J. This again indicates the relation of this pseudogap with the AFM short-range correlations which dissolve for T > J. On the other hand, the pseudogap closes also on increasing doping since it becomes barely visible at ch ∼0.11. 3. Spectral functions: Equation-of-motion approach In our analytical approach we analyze the electron Green’s function directly for projected fermionic operators [20, 21], G(k, ω) = ⟨⟨˜cks; ˜c† ks⟩⟩ω = −i Z ∞ 0 ei(ω+µ)t⟨{˜cks(t), ˜c† ks}+⟩dt, (3) 6 Pseudogap printed on October 30, 2018 −1.0 −0.5 0.0 0.5 ω/t 0.0 0.1 0.2 0.3 0.4 tN(µ+ω) T/t=0.10 0.15 0.20 0.30 −1.0 −0.5 0.0 0.5 1.0 ω/t ch=1/20 ch=2/18 (b) (a) Fig. 3. The DOS N(µ + ω) at various T ≤J for hole doping: a) ch = 1/20 and b) ch = 2/18. which is equivalent to the usual propagator within the allowed basis states of the model. In the EQM method [22] one uses relations for general correlation functions ω⟨⟨A; B⟩⟩ω = ⟨{A, B}+⟩+ ⟨⟨[A, H]; B⟩⟩ω (4) applying them to the propagator G(k, ω) [20, 21] one can represent the latter in the form G(k, ω) = α ω + µ −ζk −Σ(k, ω), (5) where α,ζk can be expressed in terms of commutators. It is", + "correlation functions ω⟨⟨A; B⟩⟩ω = ⟨{A, B}+⟩+ ⟨⟨[A, H]; B⟩⟩ω (4) applying them to the propagator G(k, ω) [20, 21] one can represent the latter in the form G(k, ω) = α ω + µ −ζk −Σ(k, ω), (5) where α,ζk can be expressed in terms of commutators. It is important to notice that the renormalization α < 1 is already the consequence of the projected basis, α = 1 N X i ⟨{˜cis, ˜c† is}+⟩= 1 2(1 + ch), (6) while ζk represents the ’free’ propagation emerging from EQM, ζk = 1 α⟨{[˜cks, H], ˜c† ks}+⟩−¯ζ = −4η1tγk −4η2t′γ′ k, (7) where ηj = α + ⟨S0·Sj⟩/α and γk = (cos kx + cos ky)/2, γ′ k = cos kx cos ky. Pseudogap printed on October 30, 2018 7 The central quantity for further consideration is the self energy Σ(k, ω) = ⟨⟨Cks; C+ ks⟩⟩irr ω /α, iCks = [˜cks, H] −ζk˜cks, (8) and only the ’irreducible’ part of the correlation function should be taken into account in the evaluation of Σ. EQM enter in the evaluation of ζk but even more important in Cks. We express the commutator in variables appropriate for a paramagnetic metallic state with ⟨Si⟩= 0, and we get [˜cks, H] = [(1 −ce 2 )ǫ0 k −Jce]˜cks + + 1 √ N X q mkq[sSz q˜ck−q,s + S∓ q ˜ck−q,−s −1 2 ˜nq˜ck−q,s], (9) where ǫ0 k = −4tγk −4t′γ′ k is the bare band energy, ˜ni = ni −ce and mkq is an effective spin-fermion coupling, mkq = 2Jγq + ǫ0 k−q. (10) One important achievement of the EQM approach is that it naturally leads to an effective coupling (not directly evident within the t-J model) between fermionic and spin degrees of freedom, which are essential for proper de- scription of low-energy physics in cuprates. Such a coupling is e.g. assumed in phenomenological models as the spin-fermion model [23, 15]. The essen- tial difference in our case is that mkq is strongly dependent on k and q just in the vicinity of most relevant ’hot’ spots. 4. Self energy 4.1. Undoped AFM and short range AFM fluctuations It is quite helpful observation that in the case of an undoped AFM our treatment of Σ and the spectral function reproduces quite successful SCBA equations [24] for the Green’s function of a hole in an AFM. If we write EQM in the coordinate space for ˜cis assuming the N´eel state as the reference nis = ±1, we get by considering only the t term, i d dt˜cis ∼−t X j n.n.i (S∓ i + S∓ j )˜cj,−s, (11) where we have also formally replaced ˜cjs = ˜cj,−sS∓ j . We note the similarity of Eq. (11) to the effective spin-holon coupling within the SCBA approach. 8 Pseudogap printed on October 30, 2018 So we can follow the procedure of the evaluation of ΣAFM(k, ω) within the SCBA in the linearized magnon theory [24] ΣAFM(k, ω) = 1 N X q M2 kqG(k −q, ω + ωq), (12) where ωq is the magnon dispersion and Mkq = 4t(uqγk−q", + "printed on October 30, 2018 So we can follow the procedure of the evaluation of ΣAFM(k, ω) within the SCBA in the linearized magnon theory [24] ΣAFM(k, ω) = 1 N X q M2 kqG(k −q, ω + ωq), (12) where ωq is the magnon dispersion and Mkq = 4t(uqγk−q + vqγk) is the holon-magnon coupling which in general strong ∝t, but vanishes near the AFM wavevector q = Q = (π, π). The advantage of the representation of EQM, Eq. (11), explicitly in spin and fermionic variables is that it allows the generalization to finite doping ch > 0. We assume that spin fluctuations remain dominant at the AFM wavevector Q with the characteristic inverse AFM correlation length κ = 1/ξAF M. The latter seems to be the case for BSCCO as well as YB2Cu3O6+x, but not for LSCO with pronounced stripe and spin-density structures [1]. For BSCCO and YBCO it is sensible to divide the spin fluc- tuations into two regimes with respect to ˜q = q −Q: a) For ˜q > κ spin fluctuations are paramagnons, they are propagating like magnons and are transverse to the local AFM short-range spin. b) For ˜q < κ spin fluctu- ations are essentially not propagating modes but critically overdamped so deviations from the long range order are essential. At finite doping case we therefore generalize (at T = 0) Eq. (19) into the paramagnon contribution, Σpm(k, ω) = 1 N X q,˜q>κ [M2 kqG−(k−q, ω+ωq)+M2 k+q,qG+(k+q, ω−ωq)], (13) where G±(k, ω) refer to the electron (ω > 0) and hole-part (ω < 0) of the propagator, respectively. We are dealing in Eq. (13) with a strong coupling theory due to t > ωq and a selfconsistent calculation of Σpm is required [21]. Also, resulting Σpm(k, ω) and A(k, ω) as are at low doping quite asymmetric with respect to ω = 0, since G−∝(1 −ch)/2 ∼1/2 while G+ ∝ch. 4.2. Longitudinal spin fluctuations At ch > 0 the electronic system is in a paramagnetic state without an AFM long-range order and besides the paramagnon excitations also the coupling to longitudinal spin fluctuations become crucial. The latter re- store the spin rotation symmetry in a paramagnet and EQM (9) introduce such a spin-symmetric coupling. Within a simplest approximation that the dynamics of fermions and spins is independent, we get Σlf(k, ω) = 1 α X q ˜m2 kq Z Z dω1dω2 π g(ω1, ω2) ˜A(k −q, ω1)χ′′(q, ω2) ω −ω1 −ω2 , (14) Pseudogap printed on October 30, 2018 9 where g(ω1, ω2) = (1/2)[th(βω1/2)+cth(βω2/2)] and χ is the dynamical spin susceptibility. Quite analogous treatment has been employed previously in the Hubbard model [25] and more recently within the spin-fermion model [15, 26]. If we want to use the analogy with the spin-fermion Hamiltonian the effective coupling parameter ˜mkq should satisfy ˜mk,q = ˜mk−q,−q which is in general not the case with the form Eq. (10), therefore we use further on instead the symmetrized coupling ˜mkq = 2Jγq + 1 2(ǫ0 k−q + ǫ0 k). (15) In contrast to previous related studies of", + "Hamiltonian the effective coupling parameter ˜mkq should satisfy ˜mk,q = ˜mk−q,−q which is in general not the case with the form Eq. (10), therefore we use further on instead the symmetrized coupling ˜mkq = 2Jγq + 1 2(ǫ0 k−q + ǫ0 k). (15) In contrast to previous related studies of spin-fermion coupling [25, 15, 26], however, ˜mkq is strongly dependent on both q and k. It is essential that in the most sensitive parts of the FS, i.e., along the AFM zone boundary (’hot’ line) where k = |Q −k|, the coupling is in fact quite modest determined solely by J and t′. Also, in the regime close to that of quasistatic χ(q, ω) the simplest and also quite satisfactory approximation is to insert for ˜A the unrenormalized A0 [15], the latter corresponding in our case to the spectral function without Σlf but with Σ = Σpm. In the present theory spin susceptibility χ(q, ω) is taken as an input. The system is close to the AFM instability, so we assume spin fluctuations of the overdamped form [23] χ′′(q, ω) ∝ ω (˜q2 + κ2)(ω2 + ω2κ). (16) Nevertheless, the appearance of the pseudogap and the form of the FS are not strongly sensitive to the particular form of χ′′(q, ω) at given character- istic κ and ωκ. 5. Pseudogap We first establish some characteristic features of the pseudogap and the development of the FS following a simplified analysis. We note that Σpm induces a large incoherent component in the spectral functions at ω ≪0 and renormalizes the effective QP band relevant to the behavior at ω ∼0 and at the FS [21]. It can also lead to a transition of a large FS into a small hole-pocket-like FS at very small ch < c∗ h ≪1. Nevertheless, the pseudogap can appear only via Σlf. Therefore we here take into account Σpm only via an effective band ǫef k . The input spectral function for Σlf is thus A0 ef(k, ω) = αZef k δ(ω + µ −ǫef k ). (17) 10 Pseudogap printed on October 30, 2018 We restrict our discussion to T = 0 and to the regime of intermediate (not too small) doping, where ǫef k defines a large FS. The simplest case is the quasi-static and single-mode approximation (QSA) which is meaningful if ωκ ≪t, κ ≪1, where we get GQSA(k, ω) = αZef k (ω −ǫef k−Q) (ω −ǫef k−Q)(ω −ǫef k ) −∆2 k . (18) The spectral function shows in this approximation two branches of E±, separated by the gap which opens along the AFM zone boundary k = kAF M and the relevant (pseudo)gap scale is ∆P G k = |∆kAF M| = Zef k 2 √rs|2J −4t′cos2kx|. (19) ∆P G k does not depend on t, but rather on smaller J and in particular t′. For t′ < 0 the gap is largest at (π, 0), consistent with experiments [7, 5]. QSA yileds a full gap corresponding, e.g., to the case of a long-range AFM state. Within the simplified effective band approach, Eq.", + "depend on t, but rather on smaller J and in particular t′. For t′ < 0 the gap is largest at (π, 0), consistent with experiments [7, 5]. QSA yileds a full gap corresponding, e.g., to the case of a long-range AFM state. Within the simplified effective band approach, Eq. (17), it is not difficult to evaluate numerically Σlf beyond the QSA, by taking ex- plicitly χ′′(q, ω), Eq. (16), with κ > 0 and ωκ ∼2Jκ. For illustration, we present results characteristic for the development of spectral functions varying two most sensitive parameters κ and µ, which both simulate the variation with doping, e.g. one can κ take in accordance with experiments [1] and numerical results on the t-J model [27] as κ ∼√ch. In Fig. 3 we present results for A(k, ω = 0) at T = 0 for a broad range of κ = 0.01 −0.6. Curves in fact display the effective FS determined by the condition G−1(kF , 0) ∼0. At the same time, intensities A(k, ω = 0) correspond to the QP weight ZF at the FS. At very small κ = 0.01 we see a small (hole-pocket) FS which follows from the QSA, Eq. (18). Already κ ∼0.05 destroys the ’shadow’ side of the pocket, i.e., the solution G−1 = 0 on the latter side disappears. On the other hand, in the gap emerge now QP solutions with very weak ZF ≪1 which reconnect the FS into a large one. We are dealing nevertheless with effectively truncated FS with well developed arcs. The effect of larger κ is essentially to increase ZF in the gapped region, in particular near (π, 0). Finally, for large κ = 0.6 corre- sponding in cuprates to the optimal doping or overdoping, ZF is essentially only weakly decreasing towards (π, 0) and the FS is well pronounced and concave as naturally expected for t′ < 0. We note in Fig. 4 that except at extreme κ = 0.01 we get a large FS, whereby in the ’underdoped’ regime κ < κ∗∼0.5 the QP weight ZF is substantial only within pronounced ’arcs’ and very small along the ’gapped’ Pseudogap printed on October 30, 2018 11 Fig. 4. Contour plot of spectral functions A(k, ω = 0) at T = 0 for various κ in one quarter of the Brillouin zone. FS where ZF ≪1. Still, such a situation corresponds to a Fermi liquid, al- though a very strange one, where QP excitations exist everywhere along the FS and hence determine the low-energy properties of the ’normal’ metallic state. It is quite remarkable to notice that in spite of ZF ≪1 the QP velocity vF is not diminished within the pseudogap. In fact it can be even enhanced, as seen in in Fig. 5 where the contour plot of A(k, ω) is shown. Again, it is well evident in Fig. 5 that QP is well defined at the FS, while it becomes fuzzy at ω ̸= 0 merging with the solutions E± k , respectively, away from the FS. The effect of large", + "5 where the contour plot of A(k, ω) is shown. Again, it is well evident in Fig. 5 that QP is well defined at the FS, while it becomes fuzzy at ω ̸= 0 merging with the solutions E± k , respectively, away from the FS. The effect of large vF in the pseudogap, which is essential for the low-T thermodynamics, can be only explained with a crucial k dependence of Σ. Assuming that k enters only via ǫ(k) we can express vF renormalization as vF vef k = (1 + ∂Σ′ ∂ǫ ) ZF Zef , ∂Σ′ ∂ǫ ω=0 ∼ ∆2 w(ωκ + w). (20) While ZF /Zef ≪1 in the same case, ∂Σ′/∂ǫ compensates or even leads to 12 Pseudogap printed on October 30, 2018 an enhancement of vF . In the case ωκw ≪∆2 we get vF vef k ∼ωκ w ∼2J vk . (21) Final vF is therefore not strongly renormalized, since 2J and vef k are of similar order. Furthermore, vF is enhanced in the parts of FS where vef k is small, in particular near (π, 0) point. The situation is thus very different from ’local’ theories where Σ(k, ω) ∼Σ(ω) and the QP velocity is governed only by ZF . In our case the ’nonlocal’ character of Σ(k, ω) is essential in order to properly describe QP within the pseudogap region. Fig. 5. Contour plot of spectral functions A(k, ω = 0) across the FS in the pseudo- gap regime. ¿From A(k, ω) we can calculate the DOS N(ω) = (2/N) P k A(k, ω). At the Fermi energy ω ∼0 and low T one can express the DOS also as (assuming the existence of QP around the FS), N(0) ∼ α 2π2 I dSFZ(kF ) v(kF ) . (22) The contribution will come mostly from FS arcs near the zone diagonal while the gapped regions near (π, 0) will contribute less due to Z(kF ) ≪1. Results in Fig. 6(a) show the development of N(0) with κ. As expected the DOS decreases with decreasing κ simulating the approach to an undoped AFM. The DOS is measured in cuprates via angle integrated photoemission spectroscopy, e.g. for LSCO in [28], as well as via the scanning tunneling Pseudogap printed on October 30, 2018 13 microscopy (STM) [29]. Our results are qualitatively consistent with these experiments (showing N(0) scaling with doping) as well as with numerical results on the t-J model [13, 14], as shown in Fig. 3. It is well possible that within photoemission experiments the matrix elements are essential, so we present in Fig. 6(a) also results weighted by the matrix element w(k) = (cos kx −cos ky)2, which originates from the interplanar hopping as proposed for the c-axis conductivity [30]. Such weighted results show a stronger dependence on doping due to enhanced influence of the pseudogap region near (π, 0), which could be even closer to experimental findings [28]. In Fig. 6(b) we show also the average Zav along the FS, as well as the QP DOS, defined as NQP = 1 2π2 I", + "a stronger dependence on doping due to enhanced influence of the pseudogap region near (π, 0), which could be even closer to experimental findings [28]. In Fig. 6(b) we show also the average Zav along the FS, as well as the QP DOS, defined as NQP = 1 2π2 I dSF v(k). (23) The decrease Zav has similar dependence at the DOS N(0). It is however quite important to notice that smaller doping (decreasing κ) leads also to a decrease of NQP. This is consistent with the observation of the pseudogap also in the specific heat in cuprates [4], since NQP ∝γ = CV /T. We note here that such a behavior is not at all evident in the vicinity of a metal- insulator transition [1]. Namely, in a Fermi liquid with (nearly constant) large FS one can drive the metal-insulator transition by Zav →0. Within an assumption of a local character Σ(ω) this would also lead to vF →0 and consequently via Eq. (23) to NQP →∞. Clearly, the essential difference in our case is that within the pseudogap regime Σ(k, ω) is nonlocal, allowing for a large (not reduced or even enhanced) velocity within the pseudogap regime, hence a simultaneous decrease of N(0) and NQP. It is also important to understand the role of finite T > 0. The most pronounced effect is on QP in the pseudogap part of the FS. The main conclusion is that weak QP peak with ZF ≪1 at T = 0, as seen e.g. in Fig. 5, is not just broadened but entirely disappears (becomes incoherent) already at very small T > Ts ≪J, e.g. Ts ∼0.02 t for the situation in Fig. 5. This can explain the puzzle that ARPES experiments in fact do not observe any QP peak near (π, 0) in the underdoped regime at T > Tc [7, 5]. 6. Conclusions In this paper we have presented our results for spectral functions and the pseudogap within the t-J model, which is the prototype model for strongly correlated electrons and for superconducting cuprates in particular. Here we first comment on the validity of the model and results in a broader perspec- tive of strange metals and materials close to the metal-insulator transition. The physics of the t-J model at lower doping levels is determined by the interplay between the magnetic exchange (dominating the undoped AFM 14 Pseudogap printed on October 30, 2018 0.0 0.2 0.4 0.6 0.8 κ 0.0 0.2 0.4 0.6 N(0) N(0) Nw(0) (a) 0.0 0.2 0.4 0.6 0.8 κ 0.0 0.1 0.2 0.3 0.4 0.5 Zav 0.0 0.5 1.0 1.5 2.0 2.5 NQP (b) NQP Zav Fig. 6. (a) DOS N(0) and weighted DOS Nw(0) vs. κ. (b) Average QP weight Zav and QP DOS NQP vs. κ. insulator) and the itinerant kinetic energy of fermions (being dominant at least in the overdoped regime). Since itinerant fermions prefer a ferromag- netic state, the quantum state at the intermediate doping is frustrated, the quantum frustration showing up in large entropy, pronounced spin fluctu- ations, non-Fermi liquid effects etc. Evidently,", + "insulator) and the itinerant kinetic energy of fermions (being dominant at least in the overdoped regime). Since itinerant fermions prefer a ferromag- netic state, the quantum state at the intermediate doping is frustrated, the quantum frustration showing up in large entropy, pronounced spin fluctu- ations, non-Fermi liquid effects etc. Evidently, this is one path towards the metal-insulator transition, but definitely not the only one possible. In this situation, fermionic and spin degrees of freedom are coupled but both active and relevant for low-energy properties. This is just the main con- tent and assumption of the presented theory for spectral functions and the pseudogap. The EQM approach to dynamical properties seem to be promising since Pseudogap printed on October 30, 2018 15 it can treat exactly the constraint which is essential for the physics of strongly correlated electrons. It has been recently also applied by present authors to the analysis of spin fluctuations and collective magnetic modes at low T. Our approximation for the self energy within the EQM approach deals with the normal paramagnetic state and treats the model as a coupled system (with derived effective coupling) of fermions with spin fluctuations, where close to the AFM ordered state both transverse and longitudinal spin fluctuations are important. Other contributions should be considered, e.g., the coupling to pairing fluctuations, in order to treat the superconducting state. In this paper we present only the results of the simplified pseudogap analysis. The results of full self-consistent treatment are qualitatively simi- lar [21]. Within the present theory the origin of the pseudogap feature is in the coupling to longitudinal spin fluctuations near the AFM wavevector Q which determine the QP properties in the ’hot’ region, i.e. near the AFM zone boundary. The pseudogap opens predominantly in the same region and its extent is dependent on J and t′ but not directly on t. Evidently the pseudogap bears a similarity to a d-wave-like dependence along the FS (for t′ < 0) being largest near the (π, 0) point. The strength of the pseudogap features depends mainly on κ. It is important to note that apart from ex- tremely small κ we are still dealing with a large FS. Still, at κ < κ∗∼0.5 parts of the FS near (π/2, π/2) remain well pronounced while the QP weight within the pseudogap part of the FS are strongly suppressed, in particular near zone corners (π, 0). The QP within the pseudogap have small weight ZF ≪1 but not dimin- ished (or even enhanced) vF , which is the effect of the nonlocal character of Σ(k, ω). A consequence is that QP within the pseudogap contribute much less to QP DOS NQP. This could be plausible explanation of a well known theoretical challenge that approaching the magnetic insulator both DOS, i.e. N(0) and NQP vanish. We presented results for T = 0, however the extention to T > 0 is straightforward. Discussing only the effect on the pseudogap, we notice that it is mainly affected by κ. So we can argue that the pseudogap should be observable for κ(ch, T)", + "i.e. N(0) and NQP vanish. We presented results for T = 0, however the extention to T > 0 is straightforward. Discussing only the effect on the pseudogap, we notice that it is mainly affected by κ. So we can argue that the pseudogap should be observable for κ(ch, T) < κ∗∼0.5. This effectively determines the crossover temperature T ∗(ch). In the region of interest κ is nearly linear in both T and ch so we would get approximately T ∗∼T ∗ 0 (1 −ch/c∗ h), (24) where T ∗ 0 ∼0.6J and c∗ h ∼0.15. As described in previous sections, several features of our theory, regard- ing the development of spectral functions, Fermi surface and pseudogap, are at least qualitatively consistent with experimental results of normal state 16 Pseudogap printed on October 30, 2018 properties in cuprates. However, further study within the present formalism is necessary in order to explore possible closer quantitative agreement with experiments as well as the emergence of superconductivity within the same model. REFERENCES [1] for a review see, e.g., M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998). [2] B. Batlogg et al., Physica C 235 - 240, 130 (1994). [3] D. C. Johnston et al., Phys. Rev. Lett. 62, 957 (1989); J. B. Torrance et al., Phys. Rev. B 40, 8872 (1989). [4] J.W. Loram, K.A. Mirza, J.R. Cooper, and W.Y. Liang, Phys. Rev. Lett. 71, 1740 (1993); J.W. Loram, J.L. Luo, J.R. Cooper, W.Y. Liang, and J.L. Tallon, Physica C 341 - 8, 831 (2000). [5] J.C. Campuzano and M. Randeria et al., in Proc. of the NATO ARW on Open Problems in Strongly Correlated Electron Systems, Eds. J. Bonˇca, P. Prelovˇsek, A. Ramˇsak, and S. Sarkar (Kluwer, Dordrecht, 2001), p. 3. [6] A. Fujimori et al., in Proc. of the NATO ARW on Open Problems in Strongly Correlated Electron Systems, Eds. J. Bonˇca, P. Prelovˇsek, A. Ramˇsak, and S. Sarkar (Kluwer, Dordrecht, 2001), p. 119. [7] D.S. Marshall et al., Phys. Rev. Lett. 76, 4841 (1996); H. Ding et al., Nature 382, 51 (1996). [8] M.R. Norman et al., Nature 392, 157 (1998). [9] K.A. Chao, J. Spa lek and A.M. Ole´s, J. Phys.C 10, L271 (1977). [10] T. Tohyama and S. Maekawa, J. Phys. Soc. Jpn. 59, 1760 (1990). [11] B. O. Wells et al., Phys. Rev. Lett. 74, 964 (1995). [12] R. Preuss, W. Hanke, C. Gr¨ober, and H.G. Evertz, Phys. Rev. Lett. 79, 1122 (1997). [13] J. Jakliˇc and P. Prelovˇsek, Phys. Rev. B 55, R7307 (1997); P. Prelovˇsek, J. Jakliˇc, and K. Bedell, Phys. Rev. B 60, 40 (1999). [14] for a review see J. Jakliˇc and P. Prelovˇsek, Adv. Phys. 49, 1 (2000). [15] A.V. Chubukov and D.K. Morr, Phys. Rep. 288, 355 (1997). [16] D. Zanchi and H.J. Schulz, Europhys. Lett. 44, 235 (1997); N. Furukawa, T.M. Rice, and M. Salmhofer, Phys. Rev. Lett. 81, 3195 (1998). [17] J. Jakliˇc and P. Prelovˇsek, Phys. Rev. B 49, 5065 (1994). [18] J. Jakliˇc and P. Prelovˇsek, Phys. Rev. Lett. 77, 892 (1996). [19] P.", + "[16] D. Zanchi and H.J. Schulz, Europhys. Lett. 44, 235 (1997); N. Furukawa, T.M. Rice, and M. Salmhofer, Phys. Rev. Lett. 81, 3195 (1998). [17] J. Jakliˇc and P. Prelovˇsek, Phys. Rev. B 49, 5065 (1994). [18] J. Jakliˇc and P. Prelovˇsek, Phys. Rev. Lett. 77, 892 (1996). [19] P. Prelovˇsek, A. Ramˇsak, and I. Sega Phys. Rev. Lett. 81, 3745 (1998). [20] P. Prelovˇsek, Z. Phys. B 103, 363 (1997). [21] P. Prelovˇsek and A. Ramˇsak, Phys. Rev. B 63, 180506 (2001); ibid. 65, 174529 (2002). Pseudogap printed on October 30, 2018 17 [22] D. N. Zubarev, Sov. Phys. Usp. 3, 320 (1960). [23] P. Monthoux, A.V. Balatsky, and D. Pines, Phys. Rev. Lett. 67, 3448 (1991); P. Monthoux and D. Pines, Phys. Rev. B 47, 6069 (1993). [24] C.L. Kane, P.A. Lee, and N. Read, Phys. Rev. B 39, 6880 (1989); G. Mart´ınez and P. Horsch, ibid. 44, 317 (1991); A. Ramˇsak and P. Prelovˇsek, ibid. 42, 10415 (1990). [25] A. Kampf and J.R. Schrieffer, Phys. Rev. B 41, 6399 (1990). [26] J. Schmalian, D. Pines, and B. Stojkovi´c, Phys. Rev. Lett. 80, 3839 (1998); Phys. Rev. B 60, 667 (1999). [27] For a review, see E. Dagotto, Rev. Mod. Phys. 66, 763 (1994). [28] A. Ino et al., Phys. Rev. Lett. 81, 2124 (1998). [29] Ch. Renner, B. Revaz, K. Kadowaki, I. Maggio-Aprile, and O. Fischer, Phys. Rev. Lett. 80, 3606 (1998). [30] L.B. Ioffe and A.J. Millis, Phys. Rev. B 58, 11631 (1998).", + "arXiv:cond-mat/0507228v4 [cond-mat.mes-hall] 7 Feb 2006 Coulomb corrections to the extrinsic spin-Hall effect of a two-dimensional electron gas E. M. Hankiewicz and G. Vignale∗ Department of Physics and Astronomy University of Missouri, Columbia, Missouri 65211, USA (Dated: November 7, 2018) We develop the microscopic theory of the extrinsic spin Hall conductivity of a two-dimensional electron gas, including skew-scattering, side-jump, and Coulomb interaction effects. We find that while the spin-Hall conductivity connected with the side-jump is independent of the strength of electron-electron interactions, the skew-scattering term is reduced by the spin-Coulomb drag, so the total spin current and the total spin-Hall conductivity are reduced for typical experimental mobilities. Further, we predict that in paramagnetic systems the spin-Coulomb drag reduces the spin accumulations in two different ways: (i) directly through the reduction of the skew-scattering contribution (ii) indirectly through the reduction of the spin diffusion length. Explicit expressions for the various contributions to the spin Hall conductivity are obtained using an exactly solvable model of the skew-scattering. PACS numbers: I. INTRODUCTION There has recently been a strong revival of interest in the phenomenon of the spin Hall effect in the context of semiconductor spintronics1. Two different forms of this phenomenon have been identified: an extrinsic spin-Hall effect2,3,4,5, which is driven by spin-orbit scattering with impurities, and an intrinsic one6,7,8, which is due to spin- orbit effects in the band structure. In both forms, the phenomenon consists of the appearance of a transverse spin current – say a z-spin current in the y-direction – when the electron gas is driven by an electric field in the x-direction. The physical manifestations of this spin current are still an active subject of study and controversy.9 However, it is now believed that the spin Hall effect should lead to transverse spin accumulation when the flow of the spin current is suppressed by an appropriate gradient of spin-dependent electrochemical potential. Recently, the spin Hall effect has been experimentally observed in Kerr rotation experiments in 3D and 2D n-doped GaAs10,11 and in a p-n junction light emitting diode (LED) realized in a two-dimensional hole system12. However, there is still a debate on the origin (extrinsic or intrinsic) of the experimentally observed spin accumulation.13,14,15,16,17 In this paper we focus exclusively on the theory of the extrinsic effect in a two-dimensional electron gas (2DEG). It has long been realized that the extrinsic spin Hall current is the sum of two contributions18. The first contribution (commonly known as “skew-scattering” mechanism19,20) arises from the asymmetry of the electron-impurity scattering in the presence of spin-orbit interactions21: electrons that are drifting in the +x direction under the action of an electric field are more likely to be scattered to the left than to the right if, say, their spin is up, while the reverse is true if their spin is down. This generates a net z-spin current in the y direction. The second contribution (the so-called “side-jump” mechanism18,22,23,24) is caused by the anomalous relationship between the physical and the canonical position operator (see Eq. (6) below). This again leads to a finite spin current in the y", + "their spin is down. This generates a net z-spin current in the y direction. The second contribution (the so-called “side-jump” mechanism18,22,23,24) is caused by the anomalous relationship between the physical and the canonical position operator (see Eq. (6) below). This again leads to a finite spin current in the y direction. The skew-scattering and side-jump contributions were widely discussed in the context of the anomalous Hall effect18,19,20,22,23,24,25,26 in magnetic materials. The skew-scattering contribution was first studied by Smit19,20 while the side-jump contribution was introduced by Berger22,23. The theory for both effects has been also discussed recently in several excellent papers, both for the extrinsic13,16,27 and the intrinsic case.28,29,30,31 In this paper in addition to the previously considered skew-scattering and side-jump contributions, we also include the Coulomb interaction effects. The main effect of interactions on the spin transport originates from the friction between spin-up and spin-down electrons moving with different drift velocities, the so called spin-Coulomb drag (SCD) effect32,33,34. We show that while the spin-Hall conductivity associated with the side-jump term is independent of the strength of electron-electron interactions, the skew scattering part is reduced by the spin-Coulomb drag, so the absolute value of the spin Hall conductivity (and hence the spin Hall current) is reduced for experimentally accessible parameters. Since the SCD has been predicted (and recently observed35) to be a rather significant contribution to the overall resistivity in high mobility electronic systems, we think it is important to include it in the description of the spin Hall effect, and we show here how this is done. Moreover, we predict that SCD in paramagnetic materials will reduce the spin accumulations through the reduction of the skew-scattering resistivity as well as the spin diffusion length. Also, we present in the Appendix a simple model for electron-impurity scattering which can be solved exactly, leading to an analytical determination of scattering rates as well as side-jump and skew-scattering contributions to the spin Hall conductivity. 2 This paper is organized as follows: in Section II the Hamiltonian and the Boltzmann equation are presented; in Section III the skew-scattering contribution to the conductivity is derived; in Section IV we use a force-balance argument to calculate the side-jump contribution; in Section V the contributions of spin Coulomb drag and spin- flip processes are included; in Section VI the spin accumulation in the presence of skew-scattering, side-jump, and electron-electron interactions is calculated. We summarize the paper in Section VII. II. HAMILTONIAN AND BOLTZMANN EQUATION We consider a strictly two-dimensional electron gas (2DEG) that lies in the x −y plane. The hamiltonian is H = H0 + Hso + Hc + HE , (1) where H0 = X i \u0014 p2 i 2m + Vei(⃗ri) \u0015 (2) is the non-interacting hamiltonian (m being the effective mass of the conduction band), including the electric potential Vei(⃗r) generated by randomly distributed impurities, Hc = 1 2 X i̸=j e2 ǫb|⃗ri −⃗rj| , (3) is the electron-electron interaction (screened by the background dielectric constant ǫb), Hso = α X i n ⃗pi × h ⃗∇iVei(⃗ri) + ⃗∇iV i ee io · ⃗σi ,", + "including the electric potential Vei(⃗r) generated by randomly distributed impurities, Hc = 1 2 X i̸=j e2 ǫb|⃗ri −⃗rj| , (3) is the electron-electron interaction (screened by the background dielectric constant ǫb), Hso = α X i n ⃗pi × h ⃗∇iVei(⃗ri) + ⃗∇iV i ee io · ⃗σi , (4) is the spin-orbit interaction (SOI) induced by the electric potential of the impurities, Vei(⃗ri), and of the other electrons ⃗V i ee = P j̸=i e2 ǫb|⃗ri−⃗rj|, and finally HE = X i n e ⃗E · ⃗ri + eα(⃗pi × ⃗E) · ⃗σi o (5) is the interaction with the external electric field ⃗E. The various spin-orbit terms appearing in the hamiltonian can all be shown to arise from a single basic fact, namely, the change in form of the physical position operator under a transformation that eliminates the coupling between the conduction band in which the electrons of interest reside, and the spin-orbit-split valence band. If we denote by ⃗ri the canonical position operator of the i-th electron, then the physical position operator is given by ⃗rphys,i = ⃗ri −α(⃗pi × ⃗σi) , (6) and correspondingly the velocity operator is: ⃗vi = −i ℏ[⃗rphys, ˆH] = ⃗pi m + 2α h ⃗∇iVei(⃗ri) + ⃗∇iV i ee + e ⃗E i × ⃗σi , (7) The spin-orbit “coupling constant” α takes into account the effective SOI induced by the valence bands (heavy holes, light holes, and split-offband) on conduction electrons in the framework of the 8-band Kane model. Within this model one finds36 α = ℏP 2 3m2e \u0014 1 E2g − 1 (Eg + ∆SO)2 \u0015 , where Eg is the gap energy between conduction and heavy/light holes bands, ∆SO is the splitting energy between heavy/light holes and split-offbands, P is the matrix element of the momentum operator between the conduction and the valence-band edges, and me is the bare electron mass. Using values of the parameters appropriate for the 3 2DEG in Al0.1Ga0.9As11 with a band mass m = 0.074me we find: αℏ= 4.4˚A2 . In this paper we treat the spin-orbit interaction to the first order in α, which is justified by the smallness of the parameter αℏ/a2 B, where aB ≃100˚A is the effective Bohr radius. Also, we consider the first order corrections from the electric potential associated with electron-electron interactions to the spin-orbit hamiltonian, i.e. the terms (αℏ/a2 B)(e2/ǫbℏvF ). Notice that the canonical positions ⃗ri and the canonical momenta ⃗pi of the particles are vectors in the x −y plane, and so is the ⃗∇operator. Therefore ⃗p×[⃗∇iVei(⃗ri)+ ⃗∇iV i ee)] is a vector in the z-direction, and the spin-orbit interaction conserves the z-component of the spin of each electron. This nice feature of our strictly 2D model allows a particularly simple analysis of the spin Hall effect without sacrificing any essential features of the spin-orbit interaction. Processes that flip the z-component of the spin will be considered separately (see Section V). We begin to exploit the conservation of σz by defining the quasi-classical one-particle distribution function fσ(⃗r,⃗k, t), i.e. the probability of finding an", + "spin Hall effect without sacrificing any essential features of the spin-orbit interaction. Processes that flip the z-component of the spin will be considered separately (see Section V). We begin to exploit the conservation of σz by defining the quasi-classical one-particle distribution function fσ(⃗r,⃗k, t), i.e. the probability of finding an electron with z-component of the spin Sz = ℏ 2σ, with σ = ±1, at position ⃗r with momentum ⃗p = ℏ⃗k at the time t. In this paper we focus on spatially homogeneous steady-state situations, in which fσ does not depend on ⃗r and t (for a discussion of non-homogeneous spin accumulation effects see Section VI). We write fσ(⃗r,⃗k, t) = f0σ(ǫk) + f1σ(⃗k) , (8) where f0σ(ǫk) is the equilibrium distribution function – a function of the free particle energy ǫk = ℏ2k2 2m – and f1σ(⃗k) is a small deviation from equilibrium induced by the application of steady electric fields ⃗Eσ (σ = ±1) which couple independently to each of the two spin components. Then to first order in ⃗Eσ the Boltzmann equation takes the form −e ⃗Eσ · ℏ⃗k m f ′ 0σ(ǫk) = ˙f1σ(⃗k)c , (9) where ˙f1σ(⃗k)c is the first-order in ⃗Eσ part of the collisional time derivative ˙fσ(⃗k)c due to different scattering processes such as electron-impurity scattering, electron-electron scattering, and spin-flip scattering. As usual, ˙fσ(⃗k)c is written as the difference of an in-scattering and and an out-scattering terms. For example, in the case of spin-conserving electron- impurity scattering one has: ˙fσ(⃗k)c,imp = − X ⃗k′ h W⃗k⃗k′σfσ(⃗k) −W⃗k′⃗kσfσ(⃗k′) i δ(eǫkσ −eǫk′σ) , (10) where W⃗k⃗k′σ is the scattering rate for a spin-σ electron to go from ⃗k to ⃗k′, and eǫkσ is the particle energy, including the additional spin-orbit interaction due the electric field ⃗Eσ. The last point is absolutely vital for a correct accounting of the “side-jump” contribution. We must use eǫkσ = ǫk + 2eαℏσ( ⃗Eσ × ˆz) · ⃗k , (11) where the second term on the right hand side differs by a factor 2 from what one would surmise from the intuitive expression ǫp + eE· rphys. Why? The reason is that the δ-function in Eq. (10) expresses the conservation of energy in a scattering process. This is a time-dependent process: therefore the correct expression for the change in position of the electron ∆rphys must be calculated as the integral of the velocity over time. From the commutator of the physical position operator with the Hamiltonian we easily find ⃗viσ = ⃗pi m + 2ασ h ⃗∇iVei(⃗ri) + ⃗∇iV i ee + e ⃗E i × ˆz , (12) where the term in the square brackets is (minus) the total force acting on the i-th electron. The time integral of this term over the duration of the collision (be it an electron-impurity or an electron-electron collision) gives the change in momentum ∆⃗p during the collision. Thus we see that the change in position is ∆⃗rphys = −2ασ∆⃗p × ˆz (this is the so-called “side-jump”): hence the change in energy is correctly given by Eq. (11). Kohn and Luttinger37, have shown that", + "electron-impurity or an electron-electron collision) gives the change in momentum ∆⃗p during the collision. Thus we see that the change in position is ∆⃗rphys = −2ασ∆⃗p × ˆz (this is the so-called “side-jump”): hence the change in energy is correctly given by Eq. (11). Kohn and Luttinger37, have shown that the above form of the collision integral is correct up to third order in the strength of the electron-impurity approximation: this is one order higher than the Born approximation and should therefore be sufficient to capture the skewdness of the scattering probability, which arises from terms beyond the Born approximation. Notice that the collision integral does not contain the tempting but ultimately incorrect “Pauli-blocking” factors 1 −fσ(⃗k′). 4 Similarly, the electron-electron contribution to the collisional derivative has the form34 ˙fσ(⃗k)c,e−e ≃− X ⃗k′⃗p⃗p′ WC(⃗kσ, ⃗p −σ;⃗k′σ, ⃗p′ −σ) n fσ(⃗k)f−σ(⃗p)[1 −fσ(⃗k′)][1 −f−σ(⃗p′)] −fσ(⃗k′)f−σ(⃗p′)[1 −fσ(⃗k)][1 −f−σ(⃗p)] o δ⃗k+⃗p,⃗k′+⃗p′δ(eǫkσ + eǫp−σ −eǫk′σ −eǫp′−σ) , (13) where WC(⃗kσ, ⃗p−σ;⃗k′σ, ⃗p′σ) is the electron-electron scattering rate from ⃗kσ, ⃗p−σ to ⃗k′σ, ⃗p′ −σ, and the Pauli factors fσ(⃗k), 1 −fσ(⃗k′) etc. ensure that the initial states are occupied and the final states empty as required by Pauli’s exclusion principle. Notice that, for our purposes, only collisions between electrons of opposite spins are relevant, since collision between same-spin electrons conserve the total momentum of each spin component. Accordingly, only the former have been retained in Eq. (13). III. SKEW-SCATTERING Let us, at first, neglect the electron-electron interaction. From the general theory developed, for instance, in Ref.38, one can easily deduce that the scattering amplitude from one impurity in two dimensions has the form f⃗k⃗k′,σ = A⃗k⃗k′ + σB⃗k⃗k′(ˆk × ˆk′)z , (14) where A⃗k⃗k′ and B⃗k⃗k′ are complex scattering amplitudes, and ˆk and ˆk′ are the unit vectors in the directions of ⃗k and ⃗k′ respectively. The second term on the right-hand side of Eq. (14) is due to the spin-orbit interaction. Squaring the scattering amplitude and multiplying by the number Ni of independent scattering centers we arrive at the following expression for the scattering rate from ⃗k to ⃗k′: W⃗k⃗k′,σ = h W s ⃗k⃗k′ + σW a ⃗k⃗k′(ˆk × ˆk′)z i δ(ǫk −ǫk′) , (15) where W s ⃗k⃗k′ = Ni \u0002 |A⃗k⃗k′|2 + |B⃗k⃗k′|2\u0003 , (16) and W a ⃗k⃗k′ = 2NiRe h A⃗k⃗k′B∗ ⃗k⃗k′ i . (17) Here and in the following ǫk ≡ℏ2k2 2m is the free particle energy. The second term in the square brackets of Eq. (15) depends on the spin (σ = +1 or −1 for up- or down-spins respectively) and on the chirality of the scattering (i.e., the sign of (ˆk × ˆk′)z). We will refer to this as the skew-scattering term. It should be noted that for a centrally symmetric scattering potential – the only case we are going to consider in this paper – W s ⃗k⃗k′ and W a ⃗k⃗k′ depend only on the magnitude of the vectors ⃗k and ⃗k′, which are equal by energy conservation, and on the angle θ between them. Furthermore, they are both symmetric under interchange of ⃗k and ⃗k′", + "are going to consider in this paper – W s ⃗k⃗k′ and W a ⃗k⃗k′ depend only on the magnitude of the vectors ⃗k and ⃗k′, which are equal by energy conservation, and on the angle θ between them. Furthermore, they are both symmetric under interchange of ⃗k and ⃗k′ – the antisymmetry of the skew-scattering being explicitly brought in by the factor (ˆk × ˆk′)z = sin θ. Thus, in the following, we will often write W s/a ⃗k⃗k′ ≡W s/a(k, θ), where W s/a(k, θ) are even functions of θ. Notice that the skew-scattering term vanishes when the scattering is treated in the second-order Born approximation38. Indeed, within this approximation A⃗k⃗k′ is purely real and B⃗k⃗k′ is purely imaginary, so W a ⃗k⃗k′ is zero. The linearized Boltzmann equation can be solved exactly under the assumption that W s ⃗k⃗k′ and W a ⃗k⃗k′ depend only on the energy ǫk = ǫk′ and on the angle θ between ⃗k and ⃗k′. The solution has the form fσ(⃗k) = f0σ(ǫk) −f ′ 0σ(ǫk)ℏ⃗k · ⃗Vσ(k) , (18) where ⃗Vσ(k) is proportional to the electric field. In view of Eq. (11) it is convenient to expand f0σ(ǫk) = f0σ(eǫkσ) −f ′ 0σ(ǫk)(eǫkσ −ǫk) . (19) 5 so that our Ansatz (18) takes the form fσ(⃗k) = f0σ(eǫkσ) −2f ′ 0σ(ǫk)eαℏσ( ⃗Eσ × ˆz) · ⃗k −f ′ 0σ(ǫk)ℏ⃗k · ⃗Vσ(k) . (20) The advantage of this form is that the “zero-order term” f0σ(eǫkσ) makes no contribution to the collision integral (10). Then, making use of Eq. (15) and discarding terms proportional to αW a (which are small since W a itself is proportional to α) we arrive at the following form for the linearized collision integral: ˙f1σ(⃗k)c,imp = − X ⃗k′ W s ⃗k⃗k′ n f1σ(⃗k) −f1σ(⃗k′) o δ(ǫk −ǫk′) −σ X ⃗k′ W a ⃗k⃗k′(ˆk × ˆk′)z n f1σ(⃗k) + f1σ(⃗k′) o δ(ǫk −ǫk′) + 2σ X ⃗k′ W s ⃗k⃗k′f ′ 0σ(ǫk)eαℏ( ⃗Eσ × ˆz) · (⃗k −⃗k′)δ(ǫk −ǫk′) , (21) where f1σ(⃗k) ≡−f ′ 0σ(ǫk)ℏ⃗k · ⃗Vσ(k) is the deviation of the distribution function from unpertubed equilibrium. At low temperature, it is not necessary to take into account the full k-dependence of ⃗Vσ(k), since the derivative of the Fermi distribution f ′ 0σ(ǫk) restricts the range of k to the vicinity of the Fermi wave vectors kF σ. Thus we replace the function Vσ(k) by a constant Vσ, and determine Vσ from the consistency condition −e X ⃗k ℏ⃗k m \" ⃗Eσ · ℏ⃗k m # f ′ 0σ(ǫk) = X ⃗k ℏ⃗k m ˙f1σ(⃗k)c,imp . (22) Substituting the collision integral from Eq. (21) on the right-hand side of this equation and moving its last term to the left hand side we arrive at enσ m ⃗Eσ −2eσ X ⃗kk′ ℏ2⃗k m h (⃗k −⃗k′)W s ⃗k⃗k′α( ⃗Eσ × ˆz)f ′ 0σ(ǫk)δ(ǫk −ǫk′) i = X ⃗k⃗k′ W s ⃗k⃗k′ ℏ2⃗k m (⃗k −⃗k′) · ⃗Vσf ′ 0σ(ǫk)δ(ǫk −ǫk′) + σ X ⃗k⃗k′ W a ⃗k⃗k′ ℏ2⃗k m (ˆk × ˆk′)z(⃗k + ⃗k′) · ⃗Vσf ′ 0σ(ǫk)δ(ǫk −ǫk′) ,", + "⃗kk′ ℏ2⃗k m h (⃗k −⃗k′)W s ⃗k⃗k′α( ⃗Eσ × ˆz)f ′ 0σ(ǫk)δ(ǫk −ǫk′) i = X ⃗k⃗k′ W s ⃗k⃗k′ ℏ2⃗k m (⃗k −⃗k′) · ⃗Vσf ′ 0σ(ǫk)δ(ǫk −ǫk′) + σ X ⃗k⃗k′ W a ⃗k⃗k′ ℏ2⃗k m (ˆk × ˆk′)z(⃗k + ⃗k′) · ⃗Vσf ′ 0σ(ǫk)δ(ǫk −ǫk′) , (23) where nσ = k2 F σ 4π is the density of σ-spin carriers and kF σ is the corresponding Fermi wave vector. The first term on the right-hand side of Eq. (23) is parallel to ⃗Vσ, while the second term is orthogonal to it. Then a simple calculation leads to the following expression for ⃗Vσ in terms of the electric field: −e m ⃗Eσ −2eασ( ⃗Eσ × ˆz) τσ = ⃗Vσ τσ + σ ⃗Vσ × ˆz τ ss σ , (24) and its inverse to first order in α is ⃗Vσ = −eτσ m \u0014 ⃗Eσ −σ τσ τ ss σ ⃗Eσ × ˆz \u0015 −2eασ( ⃗ Eσ × ˆz) , (25) where 1 τσ = − mA 4π2ℏ2ǫF σ Z ∞ 0 dǫǫf ′ 0σ(ǫ) Z 2π 0 dθ W s(ǫ, θ)(1 −cos θ) , (26) and 1 τ ss σ = − mA 4π2ℏ2ǫF σ Z ∞ 0 dǫǫf ′ 0σ(ǫ) Z 2π 0 dθ W a(ǫ, θ) sin2 θ . (27) In the above equations ǫF σ = k2 F σℏ2 2m is the Fermi energy for spin σ. In the limit of zero temperature the derivative of the Fermi function reduces to f ′ 0σ(ǫ) ≃−δ(ǫ −ǫF σ) and the above formulas simplify as follows: 1 τσ T →0 ≃ mA 4π2ℏ2 Z 2π 0 dθ W s(kF , θ)(1 −cos θ) , (28) 6 and 1 τ ss σ T →0 ≃ mA 4π2ℏ2 Z 2π 0 dθ W a(kF , θ) sin2 θ (29) 0 5 10 15 20 25 30 0.0 2.0x10 -5 4.0x10 -5 6.0x10 -5 8.0x10 -5 1.0x10 -4 1.2x10 -4 1.4x10 -4 1.6x10 -4 1.8x10 -4 W a int /(n i h/m 2 A) k 2 /a -2 k 2 F a 2 FIG. 1: Antisymmetric scattering rate W a int = R 2π 0 dθ W a(ǫ, θ) sin2 θ in units of nih/m2A (see Eq. (79)) as a function of k2 for a model circular well attractive potential V0 = −5meV and radius a = 9.45nm (described in the Appendix). We choose the parameters typical for the experimental 2DEG confined in Al0.1Ga0.9As quantum well i.e. density of electrons and impurities n2D = ni = 2.0 × 1012cm−2, m = 0.074me, and mobility ¯µ = 0.1m2/Vs. The effective spin-orbit coupling αℏ= 4.4˚A 2 in accordance with36. Fig.1 shows the antisymmetric scattering rate R 2π 0 dθ W a(ǫ, θ) sin2 θ calculated numerically using Eq. (79) for a model impurity potential Eq. (58) presented in Appendix, and for the typical experimental parameters11. IV. SPIN-HALL CURRENT AND SIDE-JUMP The quantity ⃗Vσ obtained in Eq. (25), determines the non-equilibrium distribution, according to Eq. (18). We now use this distribution to calculate the current density. In order to do this correctly, however,", + "impurity potential Eq. (58) presented in Appendix, and for the typical experimental parameters11. IV. SPIN-HALL CURRENT AND SIDE-JUMP The quantity ⃗Vσ obtained in Eq. (25), determines the non-equilibrium distribution, according to Eq. (18). We now use this distribution to calculate the current density. In order to do this correctly, however, we must remember that the spin-orbit interaction alters the relation between the velocity and the canonical momentum. The correct expression for the velocity is given in Eq. (7) and in the absence of electron-electron interactions takes the form: ⃗vi = ⃗pi m + 2α[⃗∇Vei(⃗ri) + e ⃗E] × ⃗σi . (30) The second term on the right hand side of this equation contains the net force resulting from the combined action of the external electric field and the impurity potential on the i-th electron. This force must vanish when averaged in a non-equilibrium steady-state ensemble, since the average value of the momentum must be stationary. Also, the same is true in the presence of electron-electron interactions, provided the Coulomb force is included. Thus, we arrive at the simple result ⃗jσ = −enσ⃗Vσ . (31) Combining this with Eq. (25) we obtain the complete relation between electric field and current density: ⃗jσ = nσe2τσ m \u0014 ⃗Eσ −σ τσ τ ss σ ⃗Eσ × ˆz \u0015 + 2e2ασnσ ⃗Eσ × ˆz . (32) The last term on the right-hand side of this expression is known in the literature as the “side-jump” contribution to the current density18. It comes from the use of eǫk rather than ǫk in the δ-function of conservation of energy – see 7 discussion in the paragraph following Eq. (11). Inverting Eq. (32) we obtain the formula for the electric field in terms of the current densities: ⃗Eσ = ρD σ ⃗jσ + σ[ρss σ −λσρD σ ]⃗jσ × ˆz (33) where ρD σ = m nσe2τσ is the Drude resistivity, ρss σ = m nσe2τ ss σ , λσ = 2mα τσ . Hence the resistivity tensor, written in the basis x↑, y↑, x↓, y↓, has the following form: ρ =     ρD ↑ ρss ↑−λ↑ρD ↑ 0 0 −ρss ↑+ λ↑ρD ↑ ρD ↑ 0 0 0 0 ρD ↓ −ρss ↓+ λ↓ρD ↓ 0 0 ρss ↓−λ↓ρD ↓ ρD ↓     (34) The diagonal part of the resistivity reduces to the Drude formula ρD σ = m ne2τσ as expected. The spin-orbit interaction is entirely responsible for the appearance of an off-diagonal (transverse) resistivity. The latter consists of two competing terms associated with side-jump (λσρD σ ) and skew-scattering (ρss σ ), as seen in Eq. (34). Hence our expression for the transverse resistivity is different from the expression presented in previous papers (see for example Ref.5) where only the side-jump contribution appears in the final formulas. Notice that, at this level of approximation, the resistivity is diagonal in the spin indices. V. SPIN COULOMB DRAG Up to this point we have ignored electron-electron scattering processes, as well as scattering processes that might flip the spin of the electrons. As discussed", + "contribution appears in the final formulas. Notice that, at this level of approximation, the resistivity is diagonal in the spin indices. V. SPIN COULOMB DRAG Up to this point we have ignored electron-electron scattering processes, as well as scattering processes that might flip the spin of the electrons. As discussed in Ref. (34) these processes are important because they couple the up- and down-spin components of the current density, giving rise to off-diagonal elements, ραβ ↑↓, of the resistivity tensor. The Coulomb interaction, in particular, leads to the phenomenon of the spin Coulomb drag (SCD) – a form of friction caused by the relative drift motion of spin up and spin down electrons, and the consequent transfer of momentum between them33,34,35,39. Both Coulomb and spin-flip scattering can be included in our formulation as additional contributions to the collisional derivative ˙f1σ(⃗k)c (see Ref. (34)). In particular, the Coulomb contribution is given by Eq. (13). Substituting Eq. (20) into Eq. (13) and including the first order corrections from electron-electron interactions to the spin-orbit Hamiltonian, we arrive at the following expression for the Coulomb collision integral: ˙fσ(⃗k)c,e−e ≃− 1 kBT X ⃗k′⃗p⃗p′ WC(⃗kσ, ⃗p −σ;⃗k′σ, ⃗p′ −σ)[ℏ⃗Vσ −ℏ⃗V−σ + 2eαℏσ(Eσ + E−σ) × ˆz](⃗k −⃗k′) (35) f0σ(ǫk)f0−σ(ǫp)f0σ(−ǫk′)f0−σ(−ǫp′)δ⃗k+⃗p,⃗k′+⃗p′δ(ǫkσ + ǫp−σ −ǫk′σ −ǫp′−σ) , where T is the temperature, kB is the Boltzmann constant, and we have made use of the identity f0σ(ǫk)f0−σ(ǫp)[1 − f0σ(ǫk′)][1 −f0−σ(ǫp′)] = [1 −f0σ(ǫk)][1 −f0−σ(ǫp)]f0σ(ǫk′)f0−σ(ǫp′) for ǫkσ + ǫp−σ −ǫk′σ −ǫp′−σ = 0. Eq. (35) is now inserted into the “consistency condition” (22), and the resulting sum over momenta is expressed in terms of the spin drag coefficient γ, i.e. the rate of momentum transfer between up- and down-spin electrons, according to the formula34 γ = n nσn−σ X ⃗k⃗k′⃗p⃗p′ WC(⃗kσ, ⃗p −σ;⃗k′σ, ⃗p′ −σ)(⃗k −⃗k′)2 4mkBT f0σ(ǫk)f0−σ(ǫp)f0σ(−ǫk′)f0−σ(−ǫp′) × δ⃗k+⃗p,⃗k′+⃗p′δ(ǫkσ + ǫp−σ −ǫk′σ −ǫp′−σ). (36) Then equation (24) for ⃗Vσ is modified as follows: −e m ⃗Eσ −2eασ( ⃗Eσ × ˆz) τσ = ⃗Vσ τσ + σ ⃗Vσ × ˆz τ ss σ −n−σγ nσne ⃗jσ + γ ne ⃗j−σ + 2γ n−σ n eασ( ⃗Eσ + ⃗E−σ) × ˆz + 1 τ ′σnσe ⃗j−σ, (37) The third and fourth terms on the right hand side (r.h.s.) of this equation are connected with the SCD and γ is the spin-drag coefficient calculated in Ref.39. The fifth term on r.h.s. of Eq. 37 comes from the side-jump effect in Coulomb scattering. The last term in Eq. (37) is associated with spin-flip collision processes, characterized by the spin relaxation time τ′ σ. We include it for completeness, even though spin-flip effects are expected to be small in 8 n-doped semiconductors. The current is now given by the full velocity operator of Eq. (7), but thanks to the force balance condition in the steady state it reduces again to the simple form ⃗jσ = −enσ⃗Vσ . (38) Combining (Eq. 37) with (Eq. 38) leads to the resistivity tensor, which in the basis of x↑, y↑, x↓, y↓has the form: ρ =     ρD ↑+ ρSDn↓/n↑ ρss ↑−λ↑ρD ↑+ Aγα ↑ −ρSD −ρ′", + "it reduces again to the simple form ⃗jσ = −enσ⃗Vσ . (38) Combining (Eq. 37) with (Eq. 38) leads to the resistivity tensor, which in the basis of x↑, y↑, x↓, y↓has the form: ρ =     ρD ↑+ ρSDn↓/n↑ ρss ↑−λ↑ρD ↑+ Aγα ↑ −ρSD −ρ′ ↑ Bγα ↑ −ρss ↑+ λ↑ρD ↑−Aγα ↑ ρD ↑+ ρSDn↓/n↑ −Bγα ↑ −ρSD −ρ′ ↑ −ρSD −ρ′ ↓ −Bγα ↓ ρD ↓+ ρSDn↑/n↓ −ρss ↓+ λ↓ρD ↓−Aγα ↓ Bγα ↓ −ρSD −ρ′ ↓ ρss ↓−λ↓ρD ↓+ Aγα ↓ ρD ↓+ ρSDn↑/n↓     (39) where ρSD = mγ/ne2 is the spin Coulomb drag resistivity and ρ′ σ = m/nσe2τ ′ σ (recall that λσ = 2mα τσ is a dimensionless quantity). Aγα σ and Bγα σ represent the terms of the first order in electron-electron coupling γ and in SO coupling α and are defined as follows: Aγα σ = −λσρSDn−σ/nσ + 2m∗αγ[−n−σρD σ /n + (n−σ/n −n2 −σ/nnσ)ρSD] and Bγα σ = λσρSD + 2m∗αγ[−n−σρD −σ/n + (n−σ/n −nσ/n)ρSD]. Notice that the resistivity satisfies the following symmetry relations: ρββ′ σσ = −ρβ′β σσ (40) ρββ′ σ−σ = ρβ′β −σσ (41) where upper indices β and β′ denote directions, and the lower ones spin orientations. From Eq. (39) one clearly sees that the spin Coulomb drag and spin-flip processes couple the spin components of the longitudinal resistivity. Further, the spin-Coulomb drag corrections to the spin-orbit hamiltonian (αγ corrections) result in the ρββ′ σ−σ terms, i.e. in the transverse resistivity which couples opposite spins. Also, the SCD renormalizes the longitudinal spin-diagonal components ρββ σ−σ of the resistivity, in such a way as to satisfy Galilean invariance. The spin-flip collision processes come as a phenomenological term which could have origin for example in some random magnetic field, which does not appear in the original spin-conserving Hamiltonian (Eq. (1)). The spin-flip relaxation time, τ ′ σ is given by the microscopic expression34 1 τ ′σ = − mA 8π2ℏ2ǫF −σ Z 2π 0 dθW sf σ,−σ(ǫk, θ) cos θ , (42) where W sf σ,−σ denotes the spin-flip scattering rate from spin σ to the opposite spin orientation −σ.45. Since the relaxation time for spin-flip processes τ ′ is very long,34 the SCD normally controls the coupling between the spin components, except at the very lowest temperatures (the spin drag rate γ vanishes as T 2 ln T while the spin-flip rate remains constant). Spin-flip processes will therefore be omitted henceforth. In a paramagnetic material there is symmetry between up and down spin densities, mobilities, etc. and one can easily separate the spin and charge degrees of freedom. Then combining the Eq. (37) and Eq. (38) simplifies to: ⃗Ec = ρD⃗jc + 2(ρss −λρD −λρSD)⃗js × ˆz , (43) ⃗Es = 4(ρSD + ρD)⃗js + 2(ρss −λρD −λρSD)⃗jc × ˆz , (44) where we omit spin-flip processes, and the charge/spin components of the electric field are defined as ⃗Ec = ⃗E↑+ ⃗E↓ 2 , ⃗Es = ⃗E↑−⃗E↓, and the charge and spin currents are ⃗jc = ⃗j↑+⃗j↓and ⃗js = ⃗j↑−⃗j↓ 2 ,", + "+ ρD)⃗js + 2(ρss −λρD −λρSD)⃗jc × ˆz , (44) where we omit spin-flip processes, and the charge/spin components of the electric field are defined as ⃗Ec = ⃗E↑+ ⃗E↓ 2 , ⃗Es = ⃗E↑−⃗E↓, and the charge and spin currents are ⃗jc = ⃗j↑+⃗j↓and ⃗js = ⃗j↑−⃗j↓ 2 , respectively. The spin-Coulomb drag renormalizes the longitudinal resistivity only in the spin channel. This is a consequence of the fact that the net force exerted by spin-up electrons on spin-down electrons is proportional to the difference of their drift velocities, i.e. to the spin current. Additionally, the electron-electron corrections to the spin-orbit interactions renormalize the transverse resistivity in the charge and spin channels, so the Onsager relations between spin and charge channels hold. Under the assumption that the electric field is in the x direction has the same value for spin up and spin down we see that Eqs. (43) and (44) yield the following formula for the spin current in y direction: jsz y = \u0014 ρss/2(ρD)2 1 + ρSD/ρD − λ 2ρD \u0015 Ex (45) 9 0 20 40 60 80 100 -0.001 0.000 0.001 0.002 0.003 0.004 0.005 yx sj =1m 2 /Vs yx tot =0.1m 2 /Vs =1m 2 /Vs yx tot yx ss yx ss yx [1/k ] T [K] =0.1m 2 /Vs FIG. 2: Spin Hall conductivity as a function of temperature. σsj yx(open squares), σss yx (open and close circles) are the side-jump and skew-scattering contributions, respectively, to the conductivity in the absence of electron-electron interactions, σtot yx is the total spin conductivity when electron-electron interactions are taken into account. We choose the parameters typical for the experimental 2DEG confined in Al0.1Ga0.9As quantum well i.e. density of electrons and impurities n2D = ni = 2.0×1012cm−2, m = 0.074me, and two sets of mobilities and relaxations times: ¯µ = 0.1m2/Vs, τ = 4 × 10−5ns τss = 0.02ns and ¯µ = 1m2/Vs τ = 4 × 10−4ns τss = 0.2ns. The effective spin-orbit coupling αℏ= 4.4˚A 2 in accordance with36. We used the model potential (see appendix) where an effective impurity radius a = 9.45nm, the height of attractive impurity potential V0 = −5meV for ¯µ = 0.1m2/Vs and V0 = −1.6meV for ¯µ = 1m2/Vs. The first term in the square brackets is associated with the skew-scattering, while the second is the side-jump contri- bution. Notice that the side-jump conductivity − λ 2ρD = −αne2 depends neither on the strength of disorder nor on the strength of the electron-electron interaction. In contrast, the spin-Coulomb drag reduces by factor 1 + ρSD/ρD the skew-scattering term, so the total spin Hall conductivity and the spin Hall current are reduced as well. The temperature dependence of the spin conductivities σyx = jsz y/Ex for two different mobilities and for the parameters of the recent experiment on the 2DEG in Al0.1Ga0.9As11 is presented in Fig.2. We used Eqs. (78) and (79) in the appendix to calculate the scattering rates. The procedure is as follows. First we find the scattering rate τσ(= τ−σ) from the mobility. Then using Eqs. (78)", + "for the parameters of the recent experiment on the 2DEG in Al0.1Ga0.9As11 is presented in Fig.2. We used Eqs. (78) and (79) in the appendix to calculate the scattering rates. The procedure is as follows. First we find the scattering rate τσ(= τ−σ) from the mobility. Then using Eqs. (78) and (26) we estimate the strength of impurity potential V0. Finally, we compute the skew-scattering rate by numerically evaluating Eq. (79). In general the skew-scattering rate of Eq. (26) is temperature dependent through the Fermi distribution. However, in a wide range of mobilities the impurity potential V0 is much smaller than the Fermi energy and the scattering rate does not change with energy around EF , i.e. [∂W a(ǫ, Θ)/∂ǫ]ǫ=EF ≈0 (see Fig.1). Since the temperature dependence of the skew-scattering rate comes from the energy dependence of W a(ǫ, Θ), we see that the temperature dependence of skew-scattering term is very weak. In Fig.2, σsj yx is the side-jump contribution to the conductivity in the absence of electron-electron interactions (found from the last term of Eq. 32). σss yx is the skew-scattering conductivity in the absence of electron-electron interactions (found from second term of Eq. (32)). σtot yx is the total off-diagonal conductivity modified by the spin Coulomb drag term, as given by Eq. (45). For T = 0 the total spin conductivity is the sum of two contributions: skew-scattering proportional to ρss/2ρ2 D and side-jump proportional to λ/2ρD = −αne2. For an attractive electron-impurity potential (V0 < 0) the two contributions have opposite signs, consistent with previous theoretical estimates in Refs.13,16. The ratio of the skew-scattering to side-jump terms depends on the mobility. The skew-scattering conductivity scales as µ while the side-jump is independent of mobility. As a consequence the skew-scattering conductivity dominates for high mobilities while the side-jump dominates at low mobilities. The different ratios of skew-scattering to side-jump conductivities reported in recent theoretical papers13,16 result from choosing different mobilities (see also Fig.2). The spin-Coulomb drag is the only temperature dependent contribution in this calculations: it scales as T 2 ln T for T < TF and reduces the absolute value of spin conductivity and spin current. Moreover, the reduction of spin-Hall effect by spin-Coulomb drag depends strongly on the Drude resistivity. Hence the reduction of spin-Hall conductivity will be of the order of a few percent for low mobilities (invisible change in σyx for ¯µ = 0.1m2/Vs in Fig.2) and of the order 10 of 25%-50% for high mobilities (see σyx for ¯µ = 1m2/Vs in Fig.2). Thus far, as stated in the introduction, we have only considered the extrinsic spin Hall effect. What about the intrinsic effect? In recent experiments by Sih et al.11 on a 2DEG confined in an (110) AlGaAs quantum well, three different contributions to the spin Hall conductivity are present in principle: the impurities, the linear-in-⃗k SO Rashba field, and the cubic-in-⃗k SO Dresselhaus field which is perpendicular to the Rashba field. Since in (110) quantum wells, the Dresselhaus field is in the [110] direction; we do not expect any spin-Hall current or spin-Hall", + "the spin Hall conductivity are present in principle: the impurities, the linear-in-⃗k SO Rashba field, and the cubic-in-⃗k SO Dresselhaus field which is perpendicular to the Rashba field. Since in (110) quantum wells, the Dresselhaus field is in the [110] direction; we do not expect any spin-Hall current or spin-Hall accumulation connected with this term in the plane of the quantum well. Further, it has been established that the spin-Hall conductivity in an infinite 2DEG with only linear spin-orbit interactions of Rashba or Dresselhaus type vanishes for arbitrarily weak disorder40,41,42,43. Hence we believe that the extrinsic contribution σyx ∼4 × 10−41/kΩis the dominant one in the referenced paper. Also, the theoretical estimates of extrinsic 13 and intrinsic 14 contributions in the experiments on three-dimensional n-doped GaAs 10 show that the extrinsic contribution is an order of magnitude larger than the intrinsic one. VI. SPIN ACCUMULATIONS Let us now study the influence of spin-Coulomb drag on the spin accumulation. This is relevant to the interpretation of the experiments by Sih et al.11 and Kato et al.10. We consider a very long conductor in the form of a bar of length L in the x direction and narrow width W in the y direction. A charge current flows only in the x direction. The y-components of the current jy σ, with σ =↑and ↓add up to zero everywhere and individually vanish on the edges of the system, i.e. jy σ = 0 at y = ±W/2. In order to satisfy the boundary conditions the system cannot remain homogeneous in the y-direction. A position-dependent spin density, known as spin accumulation develops across the bar, and is reflected in non-uniform chemical potentials µσ(y). In the steady state regime the spatial derivative of the spin-current in the y-direction must exactly balance the relaxation of the spin density due to spin-flip processes. This condition leads to the following equation for the spin chemical potentials44 d2[µσ(y) −µ−σ(y)] d2y = µσ(y) −µ−σ(y) L2s , (46) where Ls is the spin diffusion length. The solution of this equation is: µσ(y) −µ−σ(y) = Ce y Ls + C′e−y Ls , (47) where C, C’ are constants to be determined by the boundary conditions jy ±σ(±W/2) = 0. The effective electric field in the y- direction is eEy σ = dµσ/dy. Thus, −e(Ey ↑−Ey ↓) = d(µ↓−µ↑) dy = C′ Ls e−y/Ls −C Ls ey/Ls . (48) Using the boundary conditions at y = ±W/2 we obtain Ey ↑(±W/2) = ρyx ↑↑jx ↑+ ρyx ↑↓jx ↓ (49) Ey ↓(±W/2) = ρyx ↓↑jx ↑+ ρyx ↓↓jx ↓ (50) Making use of Eqs. (48-50) to eliminate the electric field, we obtain the following set of equations: C′ Ls e−W/2Ls −C Ls eW/2Ls = C′ Ls eW/2Ls −C Ls e−W/2Ls = −e[ρyx ↑↑jx ↑+ ρyx ↑↓jx ↓−ρyx ↓↑jx ↑−ρyx ↓↓jx ↓] , (51) which gives immediately the solution C = −C′ and C′ = −eLs[ρyx ↑↑jx ↑+ ρyx ↑↓jx ↓−ρyx ↓↑jx ↑−ρyx ↓↓jx ↓] 2 cosh(W/2Ls) . (52) Thus, the formula for the spin accumulation is µ↑(y) −µ↓(y) = eLs[ρyx ↑↑jx ↑+ ρyx ↑↓jx ↓−ρyx ↓↑jx", + "↓−ρyx ↓↑jx ↑−ρyx ↓↓jx ↓] , (51) which gives immediately the solution C = −C′ and C′ = −eLs[ρyx ↑↑jx ↑+ ρyx ↑↓jx ↓−ρyx ↓↑jx ↑−ρyx ↓↓jx ↓] 2 cosh(W/2Ls) . (52) Thus, the formula for the spin accumulation is µ↑(y) −µ↓(y) = eLs[ρyx ↑↑jx ↑+ ρyx ↑↓jx ↓−ρyx ↓↑jx ↑−ρyx ↓↓jx ↓] sinh(y/Ls) cosh(W/2Ls) . (53) 11 Finally, upon substituting the matrix elements of the resistivity from Eq. (39) we find µ↑(y) −µ↓(y) = −eLs h jx ↑(ρss ↑−λ↑ρD ↑+ Aγα ↑ + Bγα ↑) + jx ↓(ρss ↓−λ↓ρD ↓+ Aγα ↓ + Bγα ↓) i sinh(y/Ls) cosh(W/2Ls) . (54) The spin-Coulomb drag modifies the spin accumulation in three different ways: (i) the spin Coulomb drag resistivity appears directly in the terms Aγα and Bγα, defined after Eq. (39); (ii) the values of the spin components of the longitudinal current jx σ are, in general, affected by the SCD; (iii) the spin diffusion length Ls is significantly reduced by the spin Coulomb drag35 as seen from the formula34 Ls = χ0 χs Lc 1 + ρSD/ρD , (55) where χ0 is the susceptibility, χS is the spin susceptibility, and Lc is the density diffusion length. In paramagnetic materials Eq. (54) simplifies to: µ↑(y) −µ↓(y) = −2eLsjx[ρss −λρD −λρSD] sinh(y/Ls) cosh(W/2Ls) = −2eLsEx[ρss −λρD −λρSD] sinh(y/Ls) ρD cosh(W/2Ls) . (56) where jx = Ex/ρD indeed is independent of ρSD. Further the Eq. (56) at the edges of sample for L = W/2 gives: µ↑(W/2Ls) −µ↓(W/2Ls) = −2eLsjx[ρss −λρD −λρSD] tanh(W/2Ls) . (57) The three terms in the square bracket of Eq. (57) are the skew-scattering term, the side-jump contribution and the electron-electron correction. The last term reduces the spin accumulations. Additionally, the spin-Coulomb drag reduces the spin accumulation through the spin diffusion length (see Eq. (55)). However, in the limit of W ≪Ls, tanh(W/2Ls) can be approximated by W/2Ls, and the spin accumulation at the edges becomes independent of Ls. In this limit, the influence of SCD on the spin accumulation is only through the λρSD term. Notice that in the limit of infinite spin-relaxation time (Ls →∞) the spin accumulation can be obtained directly from the homogeneous formulas, Eqs. (43) and (44). For a two-dimensional electron gas confined in Al0.1Ga0.9As quantum well measured by Sih et al.11 with electron and impurity concentrations ni = n2D = 2 × 1012cm−2, mobility ¯µ=0.1m2/Vs, Ls = 1µm, τ = 4 × 10−5ns, τss = 0.02ns, αℏ= 4.4˚A and for the sample with width W = 100µm, we calculate the spin accumulation to be −1.5meV/|e| on the left edge of the sample (relative to the direction of the electric field) i.e. for W/2 = −50µm. This means that the non-equilibrium spin-density points down on the left edge of the sample and up on the right edge. It is not clear at present; whether, this sign of the spin accumulation is consistent or not with the sign of the Kerr rotation angle measured in the experiments10,11. VII. SUMMARY We have developed the microscopic theory of the extrinsic spin Hall effect taking into account the skew-scattering, side-jump", + "edge. It is not clear at present; whether, this sign of the spin accumulation is consistent or not with the sign of the Kerr rotation angle measured in the experiments10,11. VII. SUMMARY We have developed the microscopic theory of the extrinsic spin Hall effect taking into account the skew-scattering, side-jump and Coulomb interaction effects. The total spin conductivity in zero temperature is a sum of the skew- scattering and side-jump terms. The spin Coulomb drag is the only temperature dependent term, causing a quadratic- in-T reduction of the spin Hall conductivity. Further, we find that the spin Hall conductivity associated with the side-jump contribution is independent of the strength of electron-electron interactions, while the part of the spin conductivity connected with the skew scattering is reduced by the spin-Coulomb drag for experimentally accessible mobilities. Moreover, we predict that in paramagnetic systems the spin-Coulomb drag reduces the spin accumulations in two different ways: (i) directly through the reduction of the skew-scattering contribution (ii) indirectly through the reduction of the spin diffusion length. VIII. ACKNOWLEDGEMENTS This work was supported by NSF Grant No. DMR-0313681. We gratefully acknowledge valuable discussions with Shufeng Zhang, Shoucheng Zhang, Masaru Onoda and Roland Winkler. We thank Professors B. I. Halperin and E. Rashba for their criticism of an earlier version of this paper and for pointing out to us Ref.37. 12 ∗Electronic address: vignaleg@missouri.edu 1 S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Du- aghton, S. von Molnar, M. L. Rouke, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). 2 M. I. Dyakonov and V. I. Perel, Phys. Lett.A 35, 459 (1971). 3 M. I. Dyakonov and V. I. Perel, Zh. Eksp. Ter. Fiz. 13, 657 (1971). 4 J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 5 S. 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Hankiewicz, G. Vignale, and M. Flatt´e, unpublished. 44 T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993). 45 Strictly speaking spin-flip scattering also renormalizes the ordinary momentum relaxation time τσ. This small correc- tion is neglected here: it can easily be taken into account if needed. IX. APPENDIX- AN EXACTLY SOLVABLE MODEL FOR SKEW-SCATTERING We present the calculations of the scattering rates for a circular well potential of the form: V (r) = V0θ(a −r) + ¯αaLzSzδ(r −a)V0 , (58) which is attractive for V0 < 0 and repulsive for V0 > 0. The second term on the right is the spin-orbit interaction and ¯α = αℏ/a2 where α is an effective spin-orbit coupling found in 8-band Kane model and a is the impurity radius. Since the orbital angular momentum Lz = l and the spin angular momentum Sz = σ are conserved we can separate the wave function into radial and orbital parts: Ψklσ(r, θ) = Rklσ(r)eilθ (59) and the corresponding Schr¨odinger equation has the form: −ℏ2 2m \u0012 R′′ klσ + 1 r R′ klσ \u0013 + ℏ2l2 2mr2 Rklσ", + "l and the spin angular momentum Sz = σ are conserved we can separate the wave function into radial and orbital parts: Ψklσ(r, θ) = Rklσ(r)eilθ (59) and the corresponding Schr¨odinger equation has the form: −ℏ2 2m \u0012 R′′ klσ + 1 r R′ klσ \u0013 + ℏ2l2 2mr2 Rklσ + V (r)Rklσ(r) = ERklσ(r) (60) where E = ℏ2k2/2m and the prime denotes a derivative with respect to r. We now express lengths and wave vectors in units a and a−1 respectively, so r should be understood as r/a, k as ka, and, of course, a = 1 in these units. The 13 dimensionless Schr¨odinger equation is R′′ klσ + 1 r R′ klσ + \u0012 k2 −v0 −l2 r2 \u0013 Rklσ = 0, r < 1 (61) and R′′ klσ + 1 r R′ klσ + \u0012 k2 −l2 r2 \u0013 Rklσ = 0, r > 1 , (62) where v0 = 2mV0a2 ℏ2 is the dimensionless parameter which measures the height of the impurity potential barrier. The regular solution of this equation for r < 1 is Rklσ(r) = J|l|(νr), r < 1 , (63) where we have defined ν = √k2 −v0. On the other hand, the solution for r > 1 can be written as a superposition of the two independent solutions of the differential equation (62): Rklσ(r) = eiδlσ[cos δlσJ|l|(kr) −sin δlσY|l|(kr)], r > 1 (64) The matching conditions on the wave function and its derivative lead to the following equations: Rklσ(1+) = Rklσ(1−) = Rklσ(1) (65) R′ klσ(1+) −R′ klσ(1−) = ¯αlσRklσ(1)v0 (66) Substitution of Eqs. (63) and (64) to the matching conditions yields k cos δlσJ′ |l|(k) −k sin δlσY ′ |l|(k) cos δlσJ|l|(k) −sin δlσY|l|(k) = ν J′ |l|(ν) J|l|(ν) + ¯αlσv0 , (67) from which one gets the following equation for the phase shifts δlσ: cot δlσ = kY ′ |l|(k) −βlσY|l|(k) kJ′ |l|(k) −βlσJ|l|(k) (68) where βlσ = ν J′ |l|(ν) J|l|(ν) + ¯αlσv0, J|l|(k) and Y|l|(k) are the Bessel functions of the first and second kind. The wave function at large distance from the scattering center can be written as: ψklσ(r, θ) r→∞ ∼ r 2 πkr cos (kr −|l|π 2 −π 4 + δlσ)eilθeiδlσ = ψ0 klσ(r, θ) + e2iδlσ −1 √ 2πkr ei(kr−|l|π/2−π/4)eilθ (69) where ψ0 klσ(r, θ) = q 2 πkr cos (kr −|l|π 2 −π 4 )eilθ is the free wave function in the channel of angular momentum l. The scattering amplitude fσ(k, θ) is the factor multiplying the outgoing wave eikr √r in the above equation: fσ(k, θ) = ∞ X l=−∞ e2iδlσ −1 √ 2πk e−i(|l|π/2+π/4)eilθ (70) The differential cross section is accordingly given by: \u0012dσc dθ \u0013 σ = |fσ(k, θ)|2 = 1 2πk X l,l′ (e2iδlσ −1)(e−2iδl′σ −1)e−iπ/2(|l|−|l′|)ei(l−l′)θ (71) Finally we notice that the total scattering rate is related to the differential scattering cross section for a single impurity as follows: W(k, θ) = W s(k, θ) + σW a(k, θ) sin θ = ni 4π2ℏ3k m2A dσc dθ , (72) 14 where ni = Ni/A is the areal density of impurities.", + "that the total scattering rate is related to the differential scattering cross section for a single impurity as follows: W(k, θ) = W s(k, θ) + σW a(k, θ) sin θ = ni 4π2ℏ3k m2A dσc dθ , (72) 14 where ni = Ni/A is the areal density of impurities. Combining this with Eq. (71) we find W(k, θ) = ni 2π2ℏ3 m2A X l,l′ (e2iδlσ −1)(e−2iδl′σ −1)i|l′|−|l|ei(l−l′)θ, (73) To identify W s and W a we separate Eq. (73) into even and odd components with respect to the scattering angle θ, which can be easily done using the identity ei(l−l′)θ = cos[(l −l′)θ] + i sin[(l −l′)θ]. Then W s(k, θ) = ni 2π2ℏ3 m2A X l,l′ (e2iδlσ −1)(e−2iδl′σ −1)i|l′|−|l| cos[(l −l′)θ] (74) and W a(k, θ) = σni 2π2ℏ3 m2A sin θ X l,l′ (e2iδlσ −1)(e−2iδl′σ −1)i|l′|−|l|+1 sin[(l −l′)θ], (75) Making use of the identities e±2iδlσ −1 = ±2i cot δlσ∓i we rewrite the scattering rates as W s(k, θ) = ni 8π2ℏ3 m2A X l,l′ i|l′|−|l| cos[(l −l′)θ] (cot δlσ −i)(cot δl′σ + i) (76) and W a(k, θ) = σni 8π2ℏ3 m2A X l,l′ i|l′|−|l|+1 sin[(l −l′)θ] (cot δlσ −i)(cot δl′σ + i) sin θ . (77) where the phase shifts are completely determined by Eq. (68). Notice that the sums over l and l′ in Eq. (76) and Eq. (77) run from −∞to ∞and the phase shifts have the symmetries δ−l,−σ(α) = δl,σ(α) and δ−l,σ(−α) = δl,σ(α), which implies that W s(k, θ) and W a(k, θ) are invariant under spin reversal σ →−σ and W a(k, θ) changes sign with a change of sign of α, as expected. The integral over θ (Eq. 28) eliminates the majority of the terms from the sum over l and l′ and the only non-zero terms for W s(k, θ) are with l = l′ and l = l′ ± 1. This gives Z 2π 0 dθ W s(k, θ)(1 −cos θ) = ni 8π3ℏ3 m2A (X l 2 (cot δ2 lσ + 1) − X l=l′±1 i|l′|−|l| (cot δlσ −i)(cot δl′σ + i) ) . (78) One can see that W s(k, θ) is modified by spin-orbit interactions, however it has a non-zero value even if spin-orbit interactions are absent. We checked numerically that the sum over l in (Eq. 78) is convergent after taking into account a few first terms. For the skew-scattering rate W a(k, θ), the integral over θ (Eq. 29) have non-zero terms only if l = l′ ± 1. This gives: Z 2π 0 dθ W a(k, θ) sin2 θ = σni 8π3ℏ3 m2A X l=l′±1 i|l′|−|l|+1 (cot δlσ −i)(cot δl′σ + i) . (79) For very small ka the only relevant terms are l′ = 0, |l| = 1 and |l′| = 1, l = 0 which yields to: Z 2π 0 dθ W a(k, θ) sin2 θ ≃σni 8π3ℏ3 m2A \u0014 1 (cot δ1σ −i)(cot δ0σ + i) − 1 (cot δ0σ −i)(cot δ−1σ + i)+ 1 (cot δ0σ −i)(cot δ1σ + i) − 1 (cot δ−1σ −i)(cot δ0σ + i) \u0015", + "l = 0 which yields to: Z 2π 0 dθ W a(k, θ) sin2 θ ≃σni 8π3ℏ3 m2A \u0014 1 (cot δ1σ −i)(cot δ0σ + i) − 1 (cot δ0σ −i)(cot δ−1σ + i)+ 1 (cot δ0σ −i)(cot δ1σ + i) − 1 (cot δ−1σ −i)(cot δ0σ + i) \u0015 = σni 16π2ℏ3 m2A \u0014 Re 1 (cot δ1σ −i)(cot δ0σ + i) −Re 1 (cot δ−1σ −i)(cot δ0σ + i) \u0015 = = σni 16π2ℏ3 m2A \u0014 1 + cot δ0σ cot δ1σ (cot2 δ1σ + 1)(cot2 δ0σ + 1) − 1 + cot δ0σ cot δ−1σ (cot2 δ−1σ + 1)(cot2 δ0σ + 1) \u0015 , (80) independent of θ. Notice that R 2π 0 dθ W a(k, θ) sin2 θ vanishes in the absence of spin-orbit interactions, since all the phase shifts are independent of σ in that case.", + "arXiv:cond-mat/0508516v4 [cond-mat.mtrl-sci] 25 Oct 2005 Non-linear density functional theory: A direct method to calculate many-electron charge densities. Werner A. Hofer and Kriszti´an Palot´as Surface Science Research Centre, University of Liverpool, Liverpool L69 3BX, Britain Donostia International Physics Center, San Sebastian, Spain November 23, 2018 Abstract We suggest to include the density of electron charge explicitly in the electron potential of density functional theory, rather than implicitly via exchange-correlation functionals. The advantages of the approach are conceptual and numerical. Conceptually, it allows to formulate a physical principle for the development of quantum mechanical systems: it is the principle of energetic equilibrium, because the energy principle, in this case, applies not only globally, but also on a local level. The method is an order-N method, scaling linearly with the number of atoms. It is used to calculate the electronic groundstate of a metallic surface, where we find good agreement with experimental values. 1 Density functional theory A key innovation in theoretical solid state physics in the last fifty years was the reformulation of quantum mechanics in a density formalism, based on the Hohenberg-Kohn theorem [1]. Despite initial resistance, in particular from quan- tum chemists, the method has replaced previous frameworks and provides, to date, the most advanced theoretical model for the calculation of atoms, solids, and liquids. However, its implementation relies on a rather cumbersome de- tour. While the Hohenberg-Kohn theorem is formulated exclusively in terms of electron charge densities and energy functionals, calculations today are based almost exclusively on the specifications given by Kohn and Sham one year after the initial theorem was made public [2]. These Kohn-Sham equations define, how the electron interactions within a system can be split into three separate single-electron parts: a Coulomb potential UCou, giving the electron-ion attrac- tion and the electron-electron repulsion; an exchange potential UX, taking into account the Pauli exclusion principle; and a correlation potential UC, covering 1 the correlated motion of electrons in a many-electron system. Then every elec- tron in the system is described by a single-particle Schr¨odinger equation with the effective potential Ueff: Ueff = UCou + UX + UC (1) in order to determine the charge density. Within the local density approxima- tion, the exchange and correlation potentials are usually combined and thought to be depending on the local density of electron charge ρ(r): UXC(r) = UXC(ρ(r)) (2) The variational problem thus contained at every single step of the energy mini- mization cycle a solution of the single-particle Schr¨odinger equation, from which the electron eigenstates and their eigenvalues are determined. The density of charge at a given step of the cycle is calculated by adding single-electron charges: ρ(r) = N X i=1 ψi(r)ψ∗ i (r) (3) N denotes the total number of electrons in the system. While this procedure is generally successful, and implemented today in numerical methods optimized for efficiency (see e.g the ingenious way ionic and electron degrees of freedom are treated on much the same footing following a method developed by Car and Parinello [3]), it is highly inefficient in one crucial conceptual point: If, according to the", + "successful, and implemented today in numerical methods optimized for efficiency (see e.g the ingenious way ionic and electron degrees of freedom are treated on much the same footing following a method developed by Car and Parinello [3]), it is highly inefficient in one crucial conceptual point: If, according to the Hohenberg-Kohn theorem, the density of charge is the only physically relevant variable of the system, then solving the Schr¨odinger equation, setting up the eigenvectors, and computing the density of electron charge is an operation, which creates a vast amount of redundant information. Every information, pertaining to the solution of the single-particle Schr¨odinger equation and the summation of single electron charges is discarded at the end of every step in the iteration cycle. One could therefore say that more than 90% of the information created in today’s simulations is actually irrelevant. The question thus arises: Do we have to create this information at all, or can we find a more direct way to arrive at the groundstate density of charge without this cumbersome detour via the single-particle Schr¨odinger equation? From a physical point of view, this state of affairs is easy to understand, if one considers the basic innovation, quantum mechanics introduced into the original framework of mechanics. It is the property of phases and phase correlations, which, even though it is present in any many-electron system, does not find an expression in its key quantity, the density of charge. Inducing phase correlations into density functional theory thus meant to rebuild a phase coherent system by way of the original Schr¨odinger equation. The easiest way to build such a system is to determine the coherent wavefunctions of single electrons and to ensure that the states are orthogonal in Hilbert space. A more general approach would be to look at phase-correlations within a different framework. This framework, the theory of density matrices [4, 5, 6], 2 has consequently been the focus of research over a period of more than ten years. However, a framework combining the notion of phase-coherence for many-electron wavefunctions, and the basic requirement in DFT to find the groundstate density, has so far been missing. In this paper, we introduce such a direct method and show how it can be used to determine the groundstate density without any recourse to the single-particle Schr¨odinger equation. The paper was written in the spirit of orbital free density functional theory [7, 8, 9]. It is frequently pointed out by protagonists of this line of research, that an orbital free, and Bloch-state free formulation of DFT has enormous potentials for numerical improvements. The main difference to the present framework is philosophical, rather than practical. While orbital free developments are essen- tially based on the Hohenberg-Kohn theorem and attempt to find a transferable formulation for the kinetic energy functional, we have analyzed the foundation of the theory itself and sought a continuous reformulation of the fundamen- tal theory. Not least, because such a reformulation has substantial conceptual advantages, as will be shown in the following sections. The paper is organised as follows: in section II we", + "kinetic energy functional, we have analyzed the foundation of the theory itself and sought a continuous reformulation of the fundamen- tal theory. Not least, because such a reformulation has substantial conceptual advantages, as will be shown in the following sections. The paper is organised as follows: in section II we introduce the theoretical concept, based on a continuous generalisation of potentials in the many-electron Schr¨odinger equation. It is shown that the ensuing equation for the density of charge is non-linear, and that Sch¨odinger’s equation is a zero-order approxima- tion of the general relation. In section III we sketch the solution of the eigenvalue problem for a solid-state system. Finally, the new formulation is tested for metal surfaces in section IV. 2 Theoretical basis The theorem of Hohenberg and Kohn elevated the density of charge to the state of a physical variable in many-electron systems. In addition, observations by scanning tunnelling microscopes seem to confirm that the observed structures at a surface are essentially due to the distribution of electron charge, measured with a resolution of a few picometer [10]. It is then quite astonishing to find that the electron charge has to be determined by a complicated computation process from the single electron states. We show in the following, how a representation of the system, without any reference to these states, can be constructed on the basis of Green’s functions. 2.1 Green’s functions and phase correlations The Green’s function of a many-electron system can be expanded into single electron states by a spectral decomposition [11, 12]: G(r, r′, z) = N X i=1 ψi(r)ψ∗ i (r′) z −ǫi (4) Here, G(r, r′, z) is the many-electron Green’s function of the system in a local basis, N the number of electrons, and z the complex energy parameter. 3 The Green’s function will become its complex conjugate upon an exchange of the local coordinates: G(r, r′, z) = G∗(r′, r, z∗) (5) It complies with a differential equation of the Hamiltonian in a local basis: (H(r, r′) −z) G(r, r′, z) = −δ(r −r′) (6) Note that the Green’s function of the many-electron system depends only on two local coordinates r and r′. We show now that this Green’s function can be written as the product of the phase correlation functions Ψ(r) and Ψ(r′) and an energy function F(z): G(r, r′, z) = F(z)Ψ(r)Ψ∗(r′) (7) where |Ψ(r)|2 is equal to the density of charge upon a suitable choice of F(z). First, it follows from Eq. 7, that the conjugate complex of the F(z)Ψ(r)Ψ∗(r′) complies with the same relation as the Green’s function, i.e. it is equal to the transposed Green’s function: G∗(r′, r, z∗) = [F(z∗)Ψ(r′)Ψ∗(r)]∗= F(z)Ψ∗(r′)Ψ(r) = G(r, r′, z) (8) Taking the diagonal Green’s function in the limit of real eigenvalues gives: G±(r, r, E) = Ψ(r)Ψ∗(r) lim η→+0 F(E ± iη) (9) The imaginary part of the diagonal Green’s function, integrated over energy, must be equal to the charge density: ρ(r) = ∓1 π Z +∞ −∞ dEImG±(r, r, E) = ∓1 π Z +∞ −∞ dEIm lim η→+0 F(E ±", + "G±(r, r, E) = Ψ(r)Ψ∗(r) lim η→+0 F(E ± iη) (9) The imaginary part of the diagonal Green’s function, integrated over energy, must be equal to the charge density: ρ(r) = ∓1 π Z +∞ −∞ dEImG±(r, r, E) = ∓1 π Z +∞ −∞ dEIm lim η→+0 F(E ± iη)Ψ(r)Ψ∗(r) (10) If Ψ(r)Ψ∗(r) = ρ(r), the rest on the right side must be unity. This is satisfied if ∓1 π Im lim η→+0 F(E ± iη) = δ(E −E0) (11) Using the fact that lim y→+0 1 x ± iy = P \u0012 1 x \u0013 ∓iπδ(x) (12) gives us the explicit form of F(z): F(z) = 1 z −E0 (13) 4 Thus, the Green’s function can be written as the product of two local phase correlation functions[12]: G(r, r′, z) = Ψ(r)Ψ∗(r′) z −E0 (14) At this stage we have shown that the phase correlation function Ψ(r) is an exact representation of the N-electron system, as long as the system can be described by a Green’s function Eq. 4. However, it is not yet clear, whether there is a differential equation describing the evolution of Ψ(r). This equation cannot be equal to the standard Kohn-Sham equations, as this would lead to a non-local concept. If we write the Hamiltonian H(r, r′) as the sum: H(r, r′) = H(r) + H(r′) −H1(r)H∗ 1(r′), (15) and insert into Eq. 6 we get: Ψ∗(r′) [H(r) −z] Ψ(r) z −E0 + Ψ(r) [H(r′) −z] Ψ∗(r′) z −E0 − (16) − [H1(r)H∗ 1 (r′) −z] Ψ(r)Ψ∗(r′) z −E0 = −δ(r −r′) It can be seen that the second line contains a product of operators at different locations r and r′. This would mean that the phase correlation at a point r depends on events at point r′, even though there is no obvious connection. However, since Ψ depends only on r, this seems unjustified. In addition, the Hohenberg-Kohn theorem proves that if |Ψ(r)|2 is equal to the groundstate density of charge, then an energy eigenvalue E0 is defined and must be equal to the total energy. Therefore it should be possible to formulate the problem of finding the phase correlation function Ψ(r) in an eigenvalue equation. In this case the many-electron Green’s function has only one pole, in contrast to the standard spectral decomposition, where it has N poles. Here we make the only conjecture in this paper: we assume that the phase correlation function Ψ(r) is described by a suitable modification of a single coordinate Schr¨odinger equation: −ℏ2 2m∇2Ψ(r) + U(r)Ψ(r) = E0Ψ(r) (17) with a local potential U(r). That such a modification is possible and leads to correct results in simple test cases, will be shown in the following. The conjecture is supported by the derivation of an approximate single coordinate Schr¨odinger equation from the Hartree-Fock model of an N-electron system [13]. If this conjecture is justified, then we have obtained, at this point, a phase correlation function, which includes all phase correlations in the occupied range, it complies with a local equation, its corresponding eigenvalue E0 is equal to the total energy", + "from the Hartree-Fock model of an N-electron system [13]. If this conjecture is justified, then we have obtained, at this point, a phase correlation function, which includes all phase correlations in the occupied range, it complies with a local equation, its corresponding eigenvalue E0 is equal to the total energy of the system, and its square gives the electron charge density in the system. This entity looks very much like a descriptor of the physical system 5 itself. Something, which is otherwise called a many-body wavefunction. As this term might lead to confusion, since the many-body wavefunction is in standard theory described by N local coordinates (r1, r2, r3, ..., rN), we shall call it the many-electron wavefunction in the rest of the paper. An additional note concerning the relation of the framework presented in this paper to density functional theory seems necessary. In particular, since the original derivation of the Hohenberg-Kohn theorem is based on many-body wavefunctions [1]. In this respect we note that the groundstate charge density, once obtained, can be written in any representation. Thus an energy minimum, associated with a density, is enough to describe the physical content of a many- electron system. As long as the energy is a minimum, it is therefore irrelevant, how this minimum was obtained in practice. This is reflected by the most efficient methods in density functional theory, which are based on trial wave- functions, which do not diagonalize the Hamiltonian. Instead, the off-diagonal elements are minimized with the help of numerical algorithms until they vanish. 2.2 The non-linear Schr¨odinger equation So far, we have not considered the modified form of the Hamiltonian H(r). In particular the potential, which in the standard formulation is the potential of single electrons, is not suitable for our purposes. To find a local formulation of the problem, we start with the Schr¨odinger equation of the many-electron system, written as: −ℏ2 2m∇2Ψ(r) + U(r)Ψ(r) = EΨ(r) (18) Here, ∇2 denotes the Laplacian, U(r) a potential in the system, and E the total energy. Within the context of density functional theory it is generally assumed that the potential U reflects the interaction between the charge of one electron, and the potential of the environment. This entails that the potential is a sum containing the positions of all electrons in the system. While such an approach is possible, it does not reflect the previous findings. If phase correlations and charge are local quantities, then the interactions between electron charge and the potential of all other charges should also be local. The single coordinate equation 18 must therefore be recast into a relation based on densities, and not integral quantities. This provides the desired equa- tion for the many-electron wavefunction. If U(r) is e.g. the Coulomb potential of an ion with charge Z, then the potential density of electron attraction at a point (r) is given by: v(r)ρ(r) = − 1 4πε0 Ze2ρ(r) |r −Rion| v(r) = − 1 4πε0 Ze2 |r −Rion| (19) Similarly, the kinetic energy term and the total energy will change into kinetic energy density and", + "with charge Z, then the potential density of electron attraction at a point (r) is given by: v(r)ρ(r) = − 1 4πε0 Ze2ρ(r) |r −Rion| v(r) = − 1 4πε0 Ze2 |r −Rion| (19) Similarly, the kinetic energy term and the total energy will change into kinetic energy density and total energy density. The variables in the local equation will 6 be changed into: ℏ2 2m ⇒2f U(r) ⇒v(r)ρ(r) E ⇒ǫ (20) f is a system parameter, related to the volume of the system, which shall be analysed in detail further down. v(r) is the external potential. ǫ is the energy density. The equation then reads: −2f∇2Ψ(r) + v(r)ρ(r)Ψ(r) = ǫΨ(r) (21) 2.2.1 External potential The interaction between electrons and ions in a groundstate many-body problem is exclusively described by Coulomb interactions. If Ψ(r), as shown in previous sections, uniquely describes the problem, then the same should hold for the modified Schr¨odinger equation. Exchange correlation potentials, as in density functional theory, would enter the description only at the level of single electron states. Since Ψ(r) describes the total system, and not single electron state, they will not enter the present framework. However, Coulomb interactions will be modified, if one considers magnetic systems. In this case the potential v(r) will depend on the spin-state of electron charge. At present, we do not consider this case. The potential v(r) then is the Coulomb interaction potential due to electrons and ions, v(r) = e2 4πε0 \"Z d3r′ ρ(r′) |r −r′| − M X i=1 Zi |r −Ri| # (22) 2.2.2 Charge density The density of charge ρ(r) is the square of the many-electron wavefunction, or: ρ(r) = Ψ∗(r)Ψ(r) (23) Writing an equivalent equation for the complex conjugate function Ψ∗(r), we get: −2f∇2Ψ∗(r) + v(r)ρ(r)Ψ∗(r) = ǫΨ∗(r) (24) Multiplying the first of these equations by Ψ∗(r), and the second by Ψ(r), and adding the equations, we end up with: −f \u0002 Ψ∗(r)∇2Ψ(r) + Ψ(r)∇2Ψ∗(r) \u0003 + v(r)ρ(r)ρ(r) = ǫρ(r) (25) The Laplacian acting on the density is given by: ∇2 [Ψ(r)Ψ∗(r)] = \u0002 Ψ∗(r)∇2Ψ(r) + Ψ(r)∇2Ψ∗(r) \u0003 + + 2 [∇Ψ(r)] [∇Ψ∗(r)] (26) The second term on the right contains the essential phase correlations of the many-electron wavefunction. We symbolize it by Π[Ψ] to denote that it contains 7 the many-electron phase correlations. In principle there seems no straightfor- ward method to describe it in terms of charge densities alone: as the many- electron system has to be phase coherent, the density of charge alone provides insufficient information. If this were not the case, then a formulation based on density alone would be sufficient to describe the physical content of a many- electron system. From this viewpoint, the essential finding of density functional theory is not only the uniqueness of the charge distribution, but also of the groundstate energy. This energy, in turn, can only be determined by construct- ing a phase coherent system. Thus the density of charge will be described by: \u0002 −f∇2 + v(r)ρ(r) \u0003 ρ(r) + fΠ[Ψ] = ǫρ(r) Π[Ψ] = 2 [∇Ψ(r)] [∇Ψ∗(r)] (27) However, the non-linear equation describes the many-electron", + "the groundstate energy. This energy, in turn, can only be determined by construct- ing a phase coherent system. Thus the density of charge will be described by: \u0002 −f∇2 + v(r)ρ(r) \u0003 ρ(r) + fΠ[Ψ] = ǫρ(r) Π[Ψ] = 2 [∇Ψ(r)] [∇Ψ∗(r)] (27) However, the non-linear equation describes the many-electron wavefunction, provided the potential and groundstate charge density are known. It is therefore not strictly necessary to find a formulation based on density alone, as long as the nonlinear equation leads to correct many-electron wavefunctions. The derivation thus provides a set of three equations, defining the groundstate of a system: \u0002 −2f∇2 + v(r)ρ(r) \u0003 Ψ(r) = ǫΨ(r) \u0002 −f∇2 + v(r)ρ(r) \u0003 ρ(r) + fΠ[Ψ] = ǫρ(r) (28) ρ(r) = Ψ(r)Ψ∗(r) Note that this set of equations applies to the many-electron density of charge without any reference to single Kohn-Sham states. It describes the physical sys- tem by four variables: the density of charge ρ(r), the external potential v(r), which will also depend on the density, the energy density ǫ, and the many- electron wavefunction Ψ(r). The equations lead to a direct method of cal- culating the groundstate charge density by solving the nonlinear Schr¨odinger equation and minimizing the energy density ǫ of the system. The self-consistent procedure, elaborated in the following section, will be • make an initial guess for the density ρ0, • construct the potential v0, • solve the nonlinear Schr¨odinger equation for Ψ, and • construct a new charge density ρ1 = Ψ∗Ψ. The cycle is repeated until input and output charge densities are equal. The problem of finding the groundstate energy density can be formulated as the following minimization problem: ǫ = (R d3r \u0000\u0002 −f∇2 + v(r)ρ(r) \u0003 ρ(r) + fΠ[Ψ(r)] \u0001 R d3rρ(r) ) Min (29) Note that also in this case the problem cannot be formulated as a minimization problem of charge density alone. This is a consequence of basing the whole derivation on the many-electron problem and ensuring the phase coherence throughout the system. 8 2.3 Physical implications From a dimensional point of view it is interesting to analyse the meaning of the system constant f. Its dimension is that of a force: \u0002 f∇2\u0003 = \u0014 E L3 \u0015 [f] = \u0014 E L3 L2 \u0015 = \u0014E L \u0015 = N(SI) (30) Apart from the universal constants ℏand m the system constant f only depends on the volume. In this respect, it is beyond the well known Thomas-Fermi model, where the kinetic energy density is system independent. The second difference is that phase correlations are included in the picture, by solving the non-linear Schr¨odinger equation. The differences are crucial. As shown in the following sections, the model recaptures the density oscillations at metal surfaces, con- trary to the Thomas-Fermi model. That the kinetic energy density cannot be system independent, is in line with observations in quantum mechanics that the energy of a system comprising e.g. a single electron is inverse proportional to the enclosing volume. It seems therefore that the action of an external potential on the density of charge", + "That the kinetic energy density cannot be system independent, is in line with observations in quantum mechanics that the energy of a system comprising e.g. a single electron is inverse proportional to the enclosing volume. It seems therefore that the action of an external potential on the density of charge can be seen as a force acting on the wavevector, or the spatial distribution of the density. The reason for this force is that energy at a given point of the system has to be conserved. The reason that it acts on the curvature rather than the amplitude is the relation between curvature or wavevector and kinetic energy, predicted by Louis de Broglie, and experimen- tally confirmed by Davisson and Germer [14, 15]. Since ǫ was defined as the total energy divided by the volume the equation has an additional physical content. A solution of the equation with constant ǫ means that the energy density does not vary from one point of the system to another. From the viewpoint of thermodynamics, this is also the condition of a system in equilibrium. One could thus say that the specific form of the Schr¨odinger equation has one very specific physical meaning: it provides the so- lution of distributing electron charge in an external potential in such a way, that the energy density is constant. It explains, why the mathematical form of the equation is different from e.g. a differential equation in classical electrodynam- ics. There, the boundary conditions usually lead to a selection of amplitudes and frequencies in a system. Here, the frequencies need to be modified at every single point in order to achieve constant energy density values. In principle, this is still a wave-dynamical problem, but a problem modified by the ability of elec- trons to interact with external potentials and with each other. This formulation of the many-electron problem also explains, why wave mechanics, based on the Schr¨odinger equation, appears to be non-local [16]: since the main condition implicitly contained in the equation is the condition of thermal equilibrium, one cannot separate one part of the system from any other part, without contra- dicting the assumption that the system as a whole will equilibrate. Thus, wave mechanics is in fact a local theory, like every other theory known in physics, but it is applied to systems which are assumed to be connected throughout. This can clearly be seen in the mathematical form of Eqs. 28: all variables of the system, v(r),ρ(r), and ǫ are local scalars, defined at a specific point of the 9 system. These local variables cannot, however, be arbitrarily chosen, since ǫ must be a constant and f must relate to the volume of the system. One might object, at this point, that the problem is now vastly more com- plicated. After all, the eigenvalue problem is now non-linear, and it is generally impossible to compute solutions of non-linear differential equations in full gen- erality. Here, we remember the main result of the Hohenberg-Kohn theorem: the groundstate charge density is the charge density, which minimizes the total energy", + "more com- plicated. After all, the eigenvalue problem is now non-linear, and it is generally impossible to compute solutions of non-linear differential equations in full gen- erality. Here, we remember the main result of the Hohenberg-Kohn theorem: the groundstate charge density is the charge density, which minimizes the total energy and thus the energy density ǫ. This shall provide, as will be seen, a way to compute the exact solution starting from an initial guess for the charge density ρ(r), by solving a linear eigenvalue equation and adjusting the charge density from one iteration to the next. In principle, it is a very similar method to the one used today in standard DFT codes, but for two decisive advantages: 1. The charge density is computed directly without any reference to single- particle Schr¨odinger equations. 2. The charge density computed is the many-electron charge density and it is therefore completely independent of exchange-correlation functionals and density approximations. 2.4 Possible objections From a pragmatic point of view, the approach seems not too different to the standard approach, apart from the fact that the non-local density-correlations in the potential have been replaced by the charge density itself. However, there is a number of issues, related to non-linear concepts, which deserve a closer look. • If the Hohenberg-Kohn theorem is correct, then the groundstate charge density is unique. Here, we have a non-linear potential, and the ground- state is therefore potentially not unique. There are two possible answers. The first is the current state of DFT itself. After all, exchange correlation functionals include the density and its derivatives. The effective potential in today’s DFT simulations is therefore also non-linear. This, it seems, does not invalidate the results of calculations. The second is based on recent research (in Mathematics) on the behavior of non-linear Schr¨odinger equations, related to weak coupling in a many-electron systems [13]: in this case the equations still yield physically acceptable solutions, even though the existence of a solution depends on reasonable physical conditions. • The Thomas-Fermi model also attempts to calculate charge densities from a non-linear potential and a kinetic energy density. The same seems to be tried here. But we know that this model fails even to explain chemical bonding. The answer to this question lies in the treatment of the kinetic energy term, and the use of the nonlinear Schr¨odinger equation to determine many-electron charge densities. Within the Thomas-Fermi model, the prefactor of the kinetic 10 energy functional is a constant. This means, that it does not depend on the system, but is universally the same. Physically speaking, this cannot be correct. Because an electron in a very large negative potential must experience different kinetic effects than an electron in a comparatively small negative potential. In the present model the kinetic energy density is inversely proportional to the volume, and contains a term which is directly related to the many-electron phase correlations. It will be seen in the simulation of metal surfaces, that this leads to the correct behavior of electrons at a boundary. • I do not see spin-statistics involved", + "energy density is inversely proportional to the volume, and contains a term which is directly related to the many-electron phase correlations. It will be seen in the simulation of metal surfaces, that this leads to the correct behavior of electrons at a boundary. • I do not see spin-statistics involved in this approach. In standard DFT this is part of the exchange-correlation functionals. Where is it in this formulation? Here, one has to note that spin-statistics are also not involved in the formulation of the non-magnetic many-electron Schr¨odinger equation. They only arise at the level, where single electrons are thought of as individual entities. A many- electron wavefunction in a many-electron Schr¨odinger equation is a unique scalar function of location and spin. In a non-magnetic formulation only of location. So far, we have not included spin in the framework. This must be done in field mediated manner and will be the topic of future work. • Non-linearity is nothing new. It is already present in the standard tech- niques, since self-consistency cycles are essentially non-linear. The non- linearity lies therefore in the variational principle, but not in the funda- mental equations. While we do not dispute that the variation and the ensuing self-consistency cycles are non-linear, the fundamental equations, the Kohn-Sham equations, are still linear. If this were not the case, then the groundstate of a system could not be described as a superposition of N single electron states. While this, we think, is correct for a mean-field theory, it is not suitable within a many-electron framework. In this sense the described approach extends the non-linearity to the fundamental equations themselves, with the effect, that superpositions are no longer possible. The groundstate calculated with this approach cannot be decomposed into single-electron states, or only, if one uses an approximation. Before sketching the solution of the eigenvalue problem, let us consider some limiting cases, where one can show that the linear Schr¨odinger equation is in fact the zero order approximation of the general problem. 2.5 Zero order approximations The main areas, where the Schr¨odinger equation is used today, are the theory of atoms, and solid state theory. In solid state theory the most frequently employed DFT codes use periodic boundary conditions and are generally applied to systems with translational symmetry. In quantum chemistry, DFT methods usually focus on atomic orbitals and their representation by suitable basis sets. 11 Let us consider initially an atomic system, where the density of charge can be closely mimicked by an exponential function. Thus, with radial symmetry: ρ(r) = ρ0 exp (−κr) (31) The expansion of the exponential function gives: ρ(r) = ρ0 \u0012 1 −κr 1! + (κr)2 2! −(κr)3 3! + ... \u0013 (32) The zero order approximation assumes that ρ(r) = ρ0 = constant. Inserting ρ0 into Eq. 21, and multiplying by the volume V , we get: −2fV ∇2Ψ(r) + vnl(r)ρ0V Ψ(r) = EΨ(r) (33) Setting 2fV = ℏ2/2m we arrive at the conventional Schr¨odinger equation, e.g. for the hydrogen atom: \u0014 −ℏ2 2m∇2 + U(r) \u0015 Ψ(r) = EΨ(r) (34) In solid state", + "ρ0 into Eq. 21, and multiplying by the volume V , we get: −2fV ∇2Ψ(r) + vnl(r)ρ0V Ψ(r) = EΨ(r) (33) Setting 2fV = ℏ2/2m we arrive at the conventional Schr¨odinger equation, e.g. for the hydrogen atom: \u0014 −ℏ2 2m∇2 + U(r) \u0015 Ψ(r) = EΨ(r) (34) In solid state physics, systems are generally periodic. The density of charge in this case will be a periodic function. For the sake of demonstration we use a cosine, but the same argument applies to a sine or an imaginary exponent. Say, we can write the density of charge as: ρ(r) = α(1 + β cos (kr)) = α + + α · β \u0012 1 −(kr)2 2! + (kr)4 4! −(kr)6 6! + ... \u0013 (35) where k is a vector in reciprocal space, and β < 1. We set α + α · β = ρ0. Then the zero order approximation for the non-linear eigenvalue problem will read (we have again multiplied by the volume V ): −2fV ∇2Ψ(r) + vnl(r)ρ0V Ψ(r) = EΨ(r), (36) and it will again lead to the conventional Schr¨odinger equation if we set 2fV = ℏ2/2m: \u0014 −ℏ2 2m∇2 + U(r) \u0015 Ψ(r) = EΨ(r) (37) We are well aware that these examples do not constitute a proof that the non- linear eigenvalue problem is exactly equivalent to the Schr¨odinger equation. In fact, if this were the case, then the whole formulation might be considered redundant, since it is only a different way of stating what is already known. 12 2.6 Removing the paradoxes from quantum mechanics The new framework removes two of the fundamental problems in quantum me- chanics, which have been the subject of debate almost from the beginning of modern physics. It is probably not exaggerated to say that the research done on quantum paradoxes has, over the years, produced a volume of work comparable to the volume of work in standard developments. 1. The Schr¨odinger cat paradox [17]. In the standard formulation the cat is either dead, or alive, but its state can only be determined by a mea- surement. There is, fundamentally, no objective way of deciding before a measurement. 2. The Einstein-Podolsky-Rosen paradox [18]. In this case a quantum me- chanical system is thought to be split into two subsystems which increase their distance with the velocity of light. After some time two separate measurements are performed on either subsystem, and according to the standard formulation, the measurement on one subsystem should deter- mine the measurement on the second one, even though no physical inter- action is known which exceeds the speed of light. Both paradoxes rest on one fundamental feature of systems in standard quan- tum mechanics: the first, on the possibility of superimposing separate solutions of the Schr¨odinger equation to describe the state of a system; the second, on the very same feature with a slight modification concerning the times of measure- ment and the critical distance between the subsystems. The resolution of the paradoxes, within this new framework, is the easiest possible answer: the cat is dead, or it", + "describe the state of a system; the second, on the very same feature with a slight modification concerning the times of measure- ment and the critical distance between the subsystems. The resolution of the paradoxes, within this new framework, is the easiest possible answer: the cat is dead, or it is alive, independent from any measurement, because there exists at any given moment only one unique solution to the fundamental equations. This is a consequence of the non-linearity of the general framework, as already pointed out. The same applies to the second paradox, with the slight modifica- tion, that performing a measurement on one subsystem changes the state of this subsystem. While in the standard framework, this has an effect on the second subsystem, since the state of the overall system is described by a superposition, it does not affect the second subsystem in the present framework, since the fun- damental principle of equilibration would be violated. The second subsystem cannot equilibrate with the first one, because it cannot interact. From a physical point of view, this resolution of the paradoxes seems ac- tually more in keeping with a materialistic mindset. Physicists of a classical disposition have been arguing against the consequences of the mathematical framework, and in particular the feature of superposition, for almost a century. Here, we adopt the view that this consequence hinges on the linearity of the Schr¨odinger equation, and that this linearity is actually a feature enforced by the approximations used. 13 3 Iterative solution of the non-linear eigenvalue problem While some eighty years ago the non-linear problem would have been unsolv- able, this has changed significantly over the last fifty years due to the existence of powerful computers. In this respect, theory is not much different from exper- iments: a change in the equipment usually leads to a different focus in research, as phenomena, which were previously out of range, can now be analysed in a scientific manner. In particular the advent of DFT and the continuous improvement of com- puting algorithms applied to solid state physics have created an environment in theory, where the inability to solve equations analytically is no hindrance. Two of the main problems, needed to solve the non-linear problem, are already incorporated in standard DFT codes: (i) The problem of finding the charge density ρi+1 after a given iteration i. This is done by elaborate mixing schemes, which are constructed to speed up the convergency process. (ii) The problem of finding the minimum energy density. This problem is not different from the problem of finding it in standard DFT convergency cycles, where one aims at processing with the calculation along a hyper-surface, which will lead to a min- imum. In fact, if the charge density is a unique functional for a given system, then the convergence criterium can be reformulated in terms of the difference of charge in two subsequent iteration cycles: if this difference is zero, then the charge density is the true groundstate charge density. Thus the problem of finding the solution of the non-linear eigenvalue problem and the", + "given system, then the convergence criterium can be reformulated in terms of the difference of charge in two subsequent iteration cycles: if this difference is zero, then the charge density is the true groundstate charge density. Thus the problem of finding the solution of the non-linear eigenvalue problem and the groundstate charge density from: \u0002 −2f∇2 + v(r)ρ(r) \u0003 Ψ(r) = ǫΨ(r) (38) can be broken down in a number of discrete steps. Let us consider a general case; a system with a number M of ions, at the positions Ri of our coordinate system. The number of electrons N shall be: N = Z d3r Ψ∗(r)Ψ(r) = Z d3r ρ(r) = M X i=1 Zi (39) if the system is neutral and the ionic charge of atom i is given by Zi. If we neglect for the time being magnetic systems, then the potential v(r) is the sum of ionic and electronic contributions; it can be written as (we use atomic units in the rest of the paper): v(r) = − M X i=1 Zi |r −Ri| + Z d3r′ ρ(r′) |r −r′| (40) Then the potential part of the eigenvalue equation is described by (we follow the convention in DFT and write the product of external potential and charge 14 density as the effective potential veff(r)): veff(r) = v(r)ρ(r) = − M X i=1 ρ(r)Zi |r −Ri| + Z d3r′ ρ(r)ρ(r′) |r −r′| (41) The eigenvalue problem then has the form: \u0002 −2f∇2 + veff(r) \u0003 Ψ(r) = ǫΨ(r) (42) A minimization process proceeds from one charge density to the next without the calculation of Kohn-Sham eigenstates. One may wonder that the parameter f, related to the kinetic energy density, still contains the volume. This is a consequence of the rigorous application of the energy principle, not only for the system as a whole, but for every single point. If the density of charge is increased, then the potential energy will also increase. This, in turn means that the kinetic energy density will be lower. Given a number of electrons in the system, all energy components depend on the volume. The volume must therefore also show up in the kinetic energy component. 3.1 Initial charge density It is customary in DFT simulations to start a self-consistency cycle with an initial guess about the charge density distribution. In principle, this guess could involve any arbitrary choice, as long as the total number of electrons in the system is equal to N. However, most codes employ a superposition of atomic charges. As the previous section and the zero order approximation of the non- linear problem show, this is a suitable choice for the present problem. Since the charge density will ultimately be computed self-consistently, we may employ this approximation also for our initial charge distribution, setting ρ0(r) to: ρ0(r) = M X i=1 Zi X α=1 ρα,i(r −Ri) (43) where α indicates the atomic orbitals of a given atom at the position Ri. From this initial charge density the potential veff,0(r) is computed with: veff,0(r) = − M X i=1 ρ0(r)Zi |r", + "initial charge distribution, setting ρ0(r) to: ρ0(r) = M X i=1 Zi X α=1 ρα,i(r −Ri) (43) where α indicates the atomic orbitals of a given atom at the position Ri. From this initial charge density the potential veff,0(r) is computed with: veff,0(r) = − M X i=1 ρ0(r)Zi |r −Ri| + Z d3r′ ρ0(r)ρ0(r′) |r −r′| (44) 3.2 Self consistency cycle The initial potential is the input for the first iterative cycle to solve the ensuing linear density equation: \u0002 −2f∇2 + veff,0(r) \u0003 Ψ(r) = ǫΨ(r) (45) 15 The problem is a standard problem in DFT, and a large number of algorithms exists for its solution. Note that it is only necessary to calculate the lowest eigenstate, defined by ǫ1, and the corresponding real eigenvector ρ1(r). This is in marked contrast with today’s methods, where an N electron system has to be solved for N eigenvalues and eigenvectors. This means, that the size of the problem depends only on the volume of the system. If we consider that the volume usually scales linearly with the number of atoms, then the ensuing numerical problem is also of the same size. Apart from the simplification of not having to solve the Schr¨odinger equation for single Kohn-Sham states the new method scales also much better than usual codes, which scale quadratically or cubic with N. It is a true order-N method, or provides the optimum scaling with the size of the system. Today, order-N methods are based on a clever use of density matrices (see e.g [4]), or they use atomic orbitals [19], which makes them unsuitable for metallic systems. The suggested method has no limitation in this respect. Once the many-electron wavefunction is known, the charge density is given by: ρ1(r) = Ψ∗ 0(r)Ψ0(r) (46) The density ρ1(r) is used to construct a new potential, veff,1(r), by mixing the input density ρ0(r) and the output density ρ1(r) with the help of a suitable mixing algorithm, so that: ρ1(r) = M [ρ0(r), ρ1(r)] (47) Again, this is standard procedure in today’s DFT codes and a large number of mixing schemes (see e.g. [20, 21, 22]) exists to guarantee numerical stability in the self-consistency cycle. The new potential veff,1(r) is then: veff,1(r) = − M X i=1 ρ1(r)Zi |r −Ri| + Z d3r′ ρ1(r)ρ1(r′) |r −r′| (48) And the second iteration cycle consists again of solving a linear eigenvalue prob- lem described by: \u0002 −2f∇2 + veff,1(r) \u0003 Ψ(r) = ǫΨ(r) (49) A solution of the problem leads to energy eigenvalue ǫ2 and eigenvector Ψ2(r), which are used as an input for the new effective potential veff,2(r). The iteration is repeated until convergency, usually defined either by the change of eigenvalues or the change of charge density, is achieved. 3.3 Calculating the value of physical variables Once the groundstate charge density ρg(r) has been computed the many-electron wavefunction can be determined by solving the eigenvalue problem: \u0002 −2f∇2 + vg(r)ρg(r) \u0003 Ψ(r) = ǫΨ(r) (50) 16 The many-electron wavefunction may provide a direct way in the future to com- pute all physical quantities of", + "physical variables Once the groundstate charge density ρg(r) has been computed the many-electron wavefunction can be determined by solving the eigenvalue problem: \u0002 −2f∇2 + vg(r)ρg(r) \u0003 Ψ(r) = ǫΨ(r) (50) 16 The many-electron wavefunction may provide a direct way in the future to com- pute all physical quantities of interest. Since the many-electron wavefunction is the solution of the non-linear Hamiltonian, the density matrix: n(r, r′) = Ψ(r)Ψ∗(r′) (51) can be used to calculate any physical quantity A of interest via: ⟨A⟩= T r [AΨ(r)Ψ∗(r′)] = Z d3rAρ(r) (52) However, the physical quantities like kinetic energy and potential energy have been modified to obtain the general formulation. Similarly, other quantities in the Schr¨odinger equation, e.g. momenta, or electric field operators, will have to be adapted to the non-linear formulation. An alternative solution to the problem which allows to make use of the vast body of theory developed, will be to use only the many-electron groundstate charge density ρg(r), and to construct the effective potential in the standard way by: veff,dft(r) = v[ρg(r)] (53) This effective potential then allows to construct single particle solutions in the standard manner: \u001a −1 2∇2 + v[ρg(r)] \u001b ψi(r) = ǫiψi(r) (54) These solutions can be used within the standard framework, considering that the groundstate charge density completely determines the physical properties of the system. The forces on ions in a system can be calculated in the usual manner. Since the Hamiltonian of the non-linear formulation is still hermitian, the Hellman-Feynman theorem still applies [23]: Fi = −∂E ∂Ri = − Z d3rρ(r)∂v(r)ρ(r)V ∂Ri (55) From this relation the forces on atomic nuclei can be calculated in the standard manner. These forces can then be used to optimize the geometry of the system. 3.4 Summary We have shown in this section that the non-linear formulation of density func- tional theory allows to perform standard iteration procedures as used for at least three decades. It was pointed out that the main advantages of the new approach are: 1. The iterations lead to the true many-electron charge density. 2. The complexity of the calculation scales linearly with the number of atoms 17 Metal Type a [a.u.] V [a.u.3] e/cell rs Φ[eV] Cs bcc 11.6 1560.9 2 5.74 1.81 Li bcc 6.63 291.4 2 3.26 2.38 Cu fcc 6.83 318.8 44 1.20 4.40 W bcc 5.98 214.0 12 1.62 4.50 Table 1: Lattice parameters, Wigner-Seitz radii of electrons, and workfunction of typical metals. The experimental workfunctions were taken from [24], the lattice parameters from [25] 4 Numerical tests Lang and Kohn showed in the early Seventies [26, 27] that the electron charge at the surface of a metal oscillates as it approaches the surface dipole. While stan- dard DFT is able to capture this behavior, the Friedel-oscillations, the Thomas- Fermi method fails quite dramatically. It seems therefore an ideal test case for non-linear DFT. The typical density of metals can be inferred from the lattice type, the lattice constant, and the number of valence electrons, assuming that the core covers only a small region of the unit cell. The", + "Fermi method fails quite dramatically. It seems therefore an ideal test case for non-linear DFT. The typical density of metals can be inferred from the lattice type, the lattice constant, and the number of valence electrons, assuming that the core covers only a small region of the unit cell. The Wigner-Seitz radii of typical metals, as well as their experimental workfunctions, are given in Table 1. 4.1 Setup We performed self-consistent simulations of the groundstate charge density for a metallic system, varying the density of the positive background in the jellium calculation from rs = 3 to rs = 1.5a.u. The metal was modeled by a slab of 30(rs=1.5-2.0a.u.) to 40a.u.(rs = 2.5-3.0a.u.) thickness, corresponding to more than 15 layers of the metal. In standard DFT simulations this is sufficient to guarantee bulk properties in the center of the slab. In all calculations the film was embedded in two vacua, the vacuum range was 15-20 Bohr radii. This seems sufficient to mimic the surface potential barrier, it should also allow to estimate, whether the density decays exponentially into the vacuum range, as found in standard DFT simulations [26, 27], or inversely proportional with the distance from the jellium edge, as required by the asymptotic decay due to image potentials. The non-linear equation of the problem has the following form: \u0014 −1 2 d2 dz2 + vnl(z) \u0015 Ψ(z) = EΨ(z) (56) Here, the eigenvalue E is the energy value of the many-electron electron charge. With a suitably chosen non-linear potential it will be equal to the eigenvalue of the many-electron Schr¨odinger equation. The dipole potential of the surface can be calculated in the standard way by integrating the contributions from a 18 point deep inside the jellium into the vacuum region[26, 27]: vel(z) = −4π Z ∞ z dz′ Z ∞ z′ dz′′ [ρ(z′′) −ρ+(z′′)] , (57) and by using the boundary condition vel(z →∞) = 0. The nonlinear potential was obtained by multiplying the dipole potential with Vwsρ(z), where Vws is the Wigner-Seitz volume: vnl(z) = vel(z)Vwsρ(z) Vws = 4π 3 r3 s (58) The initial surface dipole was mimicked by a Fermi distibution function at both jellium edges. The initial density in the vacuum at a distance of 20a.u. was below 10−17 in units of the jellium density. The initial nonlinear potential is nearly equal to the electrostatic potential inside the slab, the only significant differences occur at the jellium edges (see Fig. 1). Conveniently, the nonlin- ear potential also has the same dimensions as the electrostatic potential. It starts deviating only, once the electron charge density in the initial setup differs substantially from the positive background. 4.2 Iterations The nonlinear equation has been solved by numerical integration. We employed the Numerov integration scheme [28]. The value of Ψn+1 on an equally spaced z-grid is the convolution of the preceding Ψ-values and the second derivative of the equations, according to: Ψn+1 = 2Ψn −Ψn−1 + h2 12 (Un+1 + 10Fn + Fn−1) 1 + h2Vn+1 12 (59) The precision of the result in one step is of the order h5,", + "an equally spaced z-grid is the convolution of the preceding Ψ-values and the second derivative of the equations, according to: Ψn+1 = 2Ψn −Ψn−1 + h2 12 (Un+1 + 10Fn + Fn−1) 1 + h2Vn+1 12 (59) The precision of the result in one step is of the order h5, which renders the method quite accurate for one dimensional applications. h is the stepsize in the integration, the derivatives Fn are given by: Fn = 2 (vnl,n −E) Ψn (60) In case of the non-linear equation Un will be zero at every gridpoint, while Vn is equal to 2 (vnl,n −E). The integration is performed twice: once starting from the left, integrating to the right, and once starting from the right and integrating left. The two functions are matched at z = 0. The parameter E, the eigenvalue of the system, determines whether they can actually be matched. The standard procedure in this case is to compute the derivatives of the two functions, Ψleft and Ψright and to change the eigenvalue E until the two functions match. Nu- merical convergency with a simple linear mixing scheme required a very high number of gridpoints (8000 gridpoints over a length of 60-80a.u.) and very low 19 mixing parameters (0.001 to 0.002) in the simulation. The difference between input and output charge density is given by the integral: Nd = 1 N Z +∞ −∞ dz|ρin(z) −ρout(z)| (61) The charge density in the iterations was converged to a value of 0.01, which seemed sufficiently precise. The eigenvalue at this level of convergency is precise to about 1meV. The difference between input and output charge density in the iterations decreases exponentially, as shown in Fig. 2. 4.3 Density oscillations In the simulations we find, in every case, an oscillation of the charge density at the jellium edge. The wavelength of the oscillation depends on the jellium density. Contrary to Lang and Kohn [26, 27] we do not find that the oscillations are damped for values of rs < 2.0a.u. The results of the simulations are shown in Fig. 3. We find the same increase of electron density at the jellium edge for every density value. The surplus charge is about 8% compared to the bulk value. The result suggests that within the non-linear framework a gradual removal of the peak and an approach of the Thomas-Fermi result in the limit of rs →2[26] does not occur. The standard DFT result for a uniform positive background cannot be extended beyond rs=3, as Lang and Kohn showed in their calculations [26, 27]. This feature of the jellium model corresponds to a failure to account for the experimental surface energies in this range. At present, we have not yet developed a method to calculate surface energies from the many-electron charge densities alone. However, it seems quite possible that results remain reasonable also in this range, considering that the surface dipole seems independent of the background charge. Analyzing the details of the density oscillations into the bulk we find a shortening of the wavelength, as the density increases. This is", + "charge densities alone. However, it seems quite possible that results remain reasonable also in this range, considering that the surface dipole seems independent of the background charge. Analyzing the details of the density oscillations into the bulk we find a shortening of the wavelength, as the density increases. This is mainly due to changes in the non-linear potential: as the potential becomes more negative, the eigenvalues do not keep up. The positive energy difference between the energy of the electron charge and the potential increases in this case, which leads to shorter wavelengths (see next sections). 4.4 Potentials and eigenvalues The peak of the charge density at the jellium edge corresponds to a trough in the attractive potential. Compared to the initial potentials (see Fig. 2) the main changes are the deepening of the dip at the barrier and the potential oscillations decaying into the bulk (see Fig. 4). Analyzing the actual process it has to be remembered that the many-electron wavefunction will be in phase throughout the film. As the two partial waves are matched at the center of the film, the actual wavelength is not completely arbitrary but reflects also the boundary conditions. From a physical point of view this is a somewhat unrealistic setup, because the waves will be reflected at the core of bulk atoms. The ensuing 20 rs [a.u.] Eigenvalue [htr] Φ [eV] Metal Φexp[eV] 3.0 -0.055 1.51 Li 2.38 2.5 -0.073 1.98 Mg 3.64 2.0 -0.102 2.80 Al 4.25 1.5 -0.160 4.35 W 4.50 Table 2: Eigenvalues and workfunctions for different densities. The workfunc- tions are in the range of 1.5 to 4.35eV, corresponding approximately to the range observed in experiments. Experimental values were taken from [24]. The deviation in the low density regime could be due to the neglect of the core region (see the text). multiple scattering effects should change the distribution of maxima and minima in a way not reproducible in a one-dimensional model. However, it is quite clear from the results that the many-electron wavefunction will have to be in-phase over a lengthscale of a few nanometer. The coherence can only be broken by electron-phonon interactions in a thermally activated environment. We would therefore conclude that long range effects are a necessary consequence of the present model. The eigenvalues obtained in the simulations are given in Table 2. They agree very well with experimental data, considering that the model is relatively simple. The only deviations are observed in the low density range rs > 3.0. This could be due either to neglecting the core volume in the calculation of the Wigner-Seitz radius, or to the fact that electron charge in a low density environment will experience the point-like and regular arrangement of the ionic cores much stronger than in a high density environment. In the first case, the actual Wigner-Seitz radius is smaller than the assumed value, in the second case the jellium model is no longer suitable. In experiments the measured values are in the range of 2.0 to 5.0eV. This corresponds to a WS radius in the range of 2.5", + "environment. In the first case, the actual Wigner-Seitz radius is smaller than the assumed value, in the second case the jellium model is no longer suitable. In experiments the measured values are in the range of 2.0 to 5.0eV. This corresponds to a WS radius in the range of 2.5 to 1.3a.u. Comparing our results for the workfunction with full potential DFT simulations [29], we note that the agreement with experiments for standard transition metals is improved. While in DFT simulations of 5d metals the typical workfunctions are well above 5eV, in case of tungsten even above 6eV, they are still in the range of 4-5eV in the present model. We may conclude that the simulations reproduce the experimental values reasonably well, and that the problems occurring in jellium simulations of high density could be due to standard density functional theory itself. Physically speaking, one would expect that a high density of electron charge makes a continuous model more applicable than a low density environment. In standard DFT jellium simulations, one observes exactly the opposite trend [26, 27]. 21 4.5 Vacuum decay Another reason for obtaining lower workfunctions than in standard DFT could be the vacuum decay of electron density. The value for rs=3.0 at a distance of 20a.u. from the surface in the fully converged system is about 10−7e/a.u.3. This is ten orders of magnitude higher than at the beginning of the self consistency loop. It is also three to four orders or magnitude higher than in standard DFT simulations. A logarithmic plot of the vacuum density reveals quite clearly, that the density does not decay exponentially in the long distance limit. Logarithmic plots of all systems are shown in Fig. 5. The curves deviate from an exponen- tial characteristic. The inset shows a fit of the density to two graphs: (i) An exponential with the decay constant equal to √ 2Φ, and a function 1/z. Both curves have the same value at the ultimate limit of the calculation, 15a.u. from the jellium edge. However, the exponential graph intersects the density curve at a discrete angle, while the 1/z curve seems to match the asymptotic density decay. The match of the density to a 1/z curve is limited to a small range of less than 5a.u. Even though the vacuum density is at least three orders of magnitude higher than in standard DFT simulations, the result is therefore not fully conclusive. We shall return to this point in future publications. 4.6 Kinetic energy functionals The potentials, energy eigenvalues, and k-values for different densities and two positions, at the center of the film, and at the peak of the surface dipole, are given in Table 3. The density, electrostatic potential, and nonlinear potential for rs = 1.5a.u. are shown in Fig. 6. The first conclusion drawn from the numerical values is that the electrostatic potential is equal to the nonlinear potential at the center of the slab. This is understandable, if one considers that the density at this point is the initial density; the product of Wigner-Seitz radius and density is in", + "The first conclusion drawn from the numerical values is that the electrostatic potential is equal to the nonlinear potential at the center of the slab. This is understandable, if one considers that the density at this point is the initial density; the product of Wigner-Seitz radius and density is in this case equal to unity. Importantly, this means that there is no correction to the electrostatic potential at this point. The situation changes at the minimum of the surface dipole potential (z = z0), about 1.5a.u. from the jellium edge. Here, the charge density deviates from the initial value, and in this case the nonlinear potential will be affected. The difference, however is small and less than 10%, as the density peak does not exceed this surplus value. Concerning the eigenvalues, we find in all cases that they are equal to the potential at the center of the film. This means, that the kinetic energy density at the center of the film must be zero. A feature, which is only possible if the curvature of the wavefunctions at this point vanishes. The density oscillations at the jellium edge therefore decay into the bulk. Since the potential at the surface dipole is lower than the eigenvalue, the wavefunctions and also the densities will oscillate. Basing the oscillations, and the relation between k-values and energies of a free electron gas in a potential vnl(z0), one would expect the k-values to comply with the dispersion relation: 1 2k2 = E −vnl(z0) (62) 22 rs vel(0) vel(z0) vnl(0) vnl(z0) E E −vnl(z0) k 1.5 -0.160 -0.290 -0.160 -0.265 -0.160 0.105 1.080 2.0 -0.102 -0.190 -0.102 -0.174 -0.102 0.072 0.910 2.5 -0.073 -0.136 -0.073 -0.124 -0.727 0.051 0.760 3.0 -0.056 -0.104 -0.056 -0.095 -0.556 0.039 0.660 Table 3: Potentials at the center of the film z = 0, at the peak of the surface dipole z = z0, eigenvalue E, difference between eigenvalue and attractive po- tential, and wavevector k. Note that the 1 2k2 is about five times larger than the difference between eigenvalue and attractive potential at z0. The difference bears on the confinement of electrons, included in the kinetic energy functional (for details, see the text). As an inspection of the numerical data shows, this is not correct. In fact, the term on the left is about five times higher than the term on the right. This means, clearly, that the kinetic energy of the nonlinear equation is not the usual single-particle or Thomas-Fermi kinetic energy. The difference becomes clear if one remembers that the fundamental change in the Schr¨odinger equation was dividing the kinetic single-electron term by the volume of the system. In a one- dimensional model, this corresponds to the lengthscale of the system. Here, we have to consider that the surface dipole is approximately 4-5a.u. wide: dividing the term on the left by 5 leads to the correct relation. Physically speaking, the change of the fundamental equation transformed the kinetic single-electron term into a kinetic energy functional. The decisive parameter of this functional is the confinement of the electrons in the", + "dipole is approximately 4-5a.u. wide: dividing the term on the left by 5 leads to the correct relation. Physically speaking, the change of the fundamental equation transformed the kinetic single-electron term into a kinetic energy functional. The decisive parameter of this functional is the confinement of the electrons in the field of the surface dipole. Regarding the energy eigenvalues it is quite clear that the energy density at the center of the slab is the same as the potential density. Approaching the jellium edge, the potential changes and the kinetic energy component changes accordingly: The system is indeed characterized by a constant energy density throughout. It is, therefore, a system in energetic equilibrium. The potential minimum between -7a.u and -3a.u. leads to an exponential decay of the wavefunction into the potential barrier and effectively screens the surface dipole from the bulk charge. This feature contains already the essence of a surface state, observed on many metal surfaces. 4.7 Summary The method was tested for a metallic surface within a continuous background model, varying the density over a range from rs=1.5 to rs = 3.0a.u. In this case we found, similarly to the results in standard DFT, that the density oscillates at the jellium edge, and that the wavelength of the oscillations depends on the jellium density. We compared the calculated eigenvalues with experimental values for the workfunctions and found good agreement, in particular in the high density range. The kinetic energy functional in the simulations includes 23 the confinement of electron charge. This has the effect that the energy density is constant throughout the system. Finally, we determined the decay of the density in the vacuum range. Here we find that the asymptotic behavior of the vacuum charge seems to be better described by a 1/z characteristic than an exponential function. 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Hofer, PhD thesis, Technische Universit¨at Wien (1999) 25 -0.4 -0.2 0 -30 -20 -10 0 10 20 30 Potential [htr] Distance [a.u.] Vacuum Vacuum Jellium rs = 2.5 rs = 1.5 rs = 2.0 rs = 3.0 0.2 0.6 1 1.4 Initial density Electrons Background Potential [eV] 0 -5.4 -10.8 Figure 1: (Color online) Setup for the calculation of metal surfaces. The jellium film is embedded in two vacuum ranges (top). The initial electrostatic potential (full graphs) inside the slab is equal to the non-linear potential (dashed graphs) as defined in Eq. 58. The potential varies with the density of the background (bottom). 26 0.01 0.1 1 0 400 800 1200 Difference Iteration Input Output Figure 2: Convergency of the iterations. The difference between input and output charge density decreases close to exponentially to its final value below 0.01. In the final iteration input and output charge density are identical (inset). 27 0.2 0.6 1 -15 -10 -5 0 5 Density [background] Distance from jellium edge [a.u.] Detail rs=3.0 rs=2.5 rs=2.0 rs=1.5 Figure 3: (Color online) Density oscillations at the jellium edge. The density shows a characteristic peak at the boundary. The maximum value of the density is nearly equivalent in all simulations and about", + "-15 -10 -5 0 5 Density [background] Distance from jellium edge [a.u.] Detail rs=3.0 rs=2.5 rs=2.0 rs=1.5 Figure 3: (Color online) Density oscillations at the jellium edge. The density shows a characteristic peak at the boundary. The maximum value of the density is nearly equivalent in all simulations and about 8% higher than the density of positive charge. The oscillations decay into the bulk, the characteristic wave- length decreases with increasing density (see detail). 28 -0.3 -0.2 -0.1 0 -10 -5 0 5 NL-Potential [htr] Distance from jellium edge [a.u.] -5.4 -2.7 NL-Potential [eV] λ/2 Figure 4: (Color online) Non-linear potential of the fully converged systems. The surface dipole corresponds to a minimum of the potential near the jellium edge. The characteristic wavelength λ decreases with increasing density. 29 1e-06 1e-04 0.01 1 0 5 10 15 20 Density [background] Distance [a.u.] exp(- 2Φ1/2z) 1/z ρ(z) Figure 5: (Color online) Density decay in the vacuum region. The logarithmic plot shows a substantial deviation from an exponential behavior in the high distance regime. 30 -0.3 -0.2 -0.1 0 0.1 -10 -5 0 5 Density and potentials [a.u.] Distance from jellium edge [a.u.] ρ vel vnl Figure 6: (Color online) Density, electrostatic potential, and nonlinear potential for rs = 1.5a.u. The potential barrier around -4a.u. screens surface electrons from bulk electrons. 31", + "SECONDARY ELECTRON EMISSION YIELD IN THE LIMIT OF LOW ELECTRON ENERGY A. N. Andronov, A. S. Smirnov, St. Petersburg State Polytechnic University, St. Petersburg, 195251 Russia I. D. Kaganovich, E. A. Startsev, and Y. Raitses, Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA V. I. Demidov, West Virginia University, USA Abstract Secondary electron emission (SEE) from solids plays an important role in many areas of science and technology.1 In recent years, there has been renewed interest in the experimental and theoretical studies of SEE. A recent study proposed that the reflectivity of very low energy electrons from solid surface approaches unity in the limit of zero electron energy2,3,4, If this was indeed the case, this effect would have profound implications on the formation of electron clouds in particle accelerators,2-4 plasma measurements with electrostatic Langmuir probes, and operation of Hall plasma thrusters for spacecraft propulsion5,6. It appears that, the proposed high electron reflectivity at low electron energies contradicts to numerous previous experimental studies of the secondary electron emission7. The goal of this note is to discuss possible causes of these contradictions. THEORETICAL DESCRIPTION OF SECONDARY ELECTRON EMISSION IN THE LIMIT OF LOW ENERGY Authors of the Refs. [2-4] suggest that the theoretical description of the elastic backscattering of low-energy electrons at solid surface could be adequately described by a very simplified one-dimensional model of the quantum reflection of electron plane waves incident on an abrupt potential step of height, eVi of the internal potential. For example, in R. Cimino’s work3 it was assumed that eVi =150 eV, which is unrealistically large number for typical internal potential of order several electronvolts. Such a large number was assumed to explain experimental result that the secondary electron emission yield (SEEY) tends to unity as energy approaches 0 at electron energies below 10eV. It was assumed that electrons are reflected back only at the surface potential barrier and do not penetrate into the solid. Such description is oversimplified for several reasons. First, electron interaction with the real surface of a solid target cannot be described by a potential barrier with a sharp step. At distances x greater than the interatomic distance in solids when the metal surface can be considered as perfectly smooth and perfectly conducting, electron experiences an image force, which is equal to e2/4x (the so-called Schottky effect8). At shorter distances the metal surface cannot be considered as perfectly smooth and the work function is determined by the dipole moments of surface atoms, preventing the exit of electrons into vacuum. In this region the potential is a nearly-constant and equal to the inner potential of the solid. Quantum-mechanical reflectivity, R, of slow electrons at the barrier of such a form is taken into account in the Richardson law for the thermionic emission current density9, J: J=A0T2D exp (-φ/kT), where A0 is the Sommerfeld constant=120.4A/cm2K2, D=1-R is transparency of the barrier, and φ is the local work function of the sample. It is shown experimentally that for thermal electrons (energy in vacuum less than 0.1 eV) the average coefficient of reflection at the surface is less", + "J=A0T2D exp (-φ/kT), where A0 is the Sommerfeld constant=120.4A/cm2K2, D=1-R is transparency of the barrier, and φ is the local work function of the sample. It is shown experimentally that for thermal electrons (energy in vacuum less than 0.1 eV) the average coefficient of reflection at the surface is less than 10 percent, and in most cases much less. According to the laws of quantum mechanics, the reflection does not depend on the direction of electron velocity. Therefore, coefficient of reflection at the barrier must be the same for both, the thermal electrons that are escaping from the solid, and for the incident primaries. the bottom of the conduction band the vacuum level Internal potencial eVi EF φ Ek ES Figure 1: The energy level diagram. Additional contribution to the elastic reflection is caused by electrons scattered inside a solid target. Incident primary electrons are accelerated by the surface potential barrier to the value Es=Ek+EF+φ (see Fig.1) and then they penetrate into a solid target to a depth of 50-100 Å 10,11. In that process electrons lose their energy and produce secondary electrons, and are also elastically scattered by the atoms of a solid. This scattering (so called s- scattering) is isotropic and its cross section is determined by the atomic number of scattering atoms and does not depend on energy when the electron wavelength is much greater than inter-atomic distance12. A half of the elastically scattered electrons are scattered by an angle of more than π/2 (back-scattered electrons). These electrons also undergo the elastic and inelastic scattering on their way back to the surface and only a small fraction reaches the surface with an initial energy Es ≥ eVi. When an electron passes through the surface potential barrier in vacuum only, the normal to the surface component of the electron momentum changes and the parallel to the surface component of the electron momentum does not change. Therefore, if electron has small kinetic energy in vacuum, Ek, this electron cannot overcome the potential barrier after scattering on atoms unless it scattered exactly backward 180 degrees. In the limit Ek → 0, the contribution of these back scattered electrons to the total elastic reflection tends to zero (not to unity). Thus, this process also does not increase the elastic reflection to full 100 percent eflection with kinetic energy of the electrons in vacuum decreasing to zero. EXPERIEMTNAL CONSIDERATIONS AND DIFFICULTIES OF MEASUREING THE SECONDARY ELECTRON EMISSION YIELD IN THE LIMIT OF LOW ENERGY Experimental measurements at low incident electron energy, below 2eV, are extremely challenging. It is very difficult to produce collimated aligned electron beam at such low energy. References 7 and 18 measured reflection coefficient down to 3 eV making use of low- energy electron beam. They observed reflection coefficient below 10 percent for clean targets and 40% for contaminated targets with some absorbed gas on surface. Because it is very difficult to produce electron beam with such low energy, common approach to study secondary electron yield is to use an electron gun at a given fixed energy and slow down electrons", + "percent for clean targets and 40% for contaminated targets with some absorbed gas on surface. Because it is very difficult to produce electron beam with such low energy, common approach to study secondary electron yield is to use an electron gun at a given fixed energy and slow down electrons by applying retarding potential to the target. But “when the incident energy is decreased by an increase of the negative bias of the sample holder instead of a decrease of the nominal beam energy E, such a bias leads to constant potential surfaces in the vacuum gap on which the incident electrons may be totally reflected and then collected without any contact with the sample surface”. Making use of such approach authors of Ref. [13] reported the reflection coefficient 40% whereas authors of Ref. [14] reported reflection coefficient 100%. The most recent detailed review of the latest research in the field of reflectivity of very low energy electrons from solid surfaces Ref. [1] refers to the Ref. [2] as the main experimental evidence of the proposed hypothesis. The examination of description of experiment conducted by these authors shows that they used experimental setup that is commonly used to measure a contact potential difference between cathode and a target making use of an electron beam (also so-called the Anderson method 15,16,17]). In this method, the contact potential difference between cathode and a target is determined by the condition when the current between cathode and anode (target) is zero. Therefore, in order to achieve that the electron beam with low energy reaches the target, the voltage on the target should be carefully chosen to be equal the beam energy minus the contact potential difference between the cathode and the target. Without taken into account the effect of the contact potential difference between the cathode and the target on beam energy, the incident beam electron may reflect from the retarding potential in the vacuum gap without reaching the target. This reflection from the retarding potential in the vacuum gap then will be inaccurately interpreted as 100% elastic reflection from the surface. It is difficult to access what was the exact reason for artificial 100% reflection reported in the experiments of Ref.[2]. However, comparing the data of Ref.[2] with the data confidently obtained in numerous previous careful measurements reported in Refs.[19,20,21] shows big differences for electron energies in the range 5-10 eV. For example, Fig.2 depicts comparison of the results under discussion with previously published data for copper targets. From Fig.2 it is evident that there is sufficiently 0 5 10 15 20 25 30 35 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 2 1 δ Primary energy (eV) Figure 2: Total secondary electron emission yield of Cu at low electron primary energy Ep. 1. Initial part of Fig. 2 taken from the letter [3] for fully scrubbed Cu (T=10 K). 2. Experimental data for Cu from Ref. [18] after applying heat in vacuum (taken at room temperature). good agreement between two data sets for Ep>10 eV, but there are drastic differences", + "energy Ep. 1. Initial part of Fig. 2 taken from the letter [3] for fully scrubbed Cu (T=10 K). 2. Experimental data for Cu from Ref. [18] after applying heat in vacuum (taken at room temperature). good agreement between two data sets for Ep>10 eV, but there are drastic differences for Ep<10 eV. However, the data of Ref. [3] were taken for a cryogenically cooled target whereas data of Ref. [18] were taken at normal conditions. As follows from the theoretical description that should not make a big difference. In summary, we presented ample evidence that numerous previous measurements in the low energy range shows reflection coefficient of about 7% in the range of few electron volts. We also presented compelling arguments that Refs. [2-4] provide contradictory account of SEE at low energies when compared with other numerous previous measurements in the low energy range given by authors of Refs. [7, 18-21]. In addition, there are straightforward theoretical considerations that support the claim that the reflection coefficient should remain small even in the limit of very low electron energy. ACKNOWLEDGMENT This work was supported by the U.S. Department of Energy and the U.S. Air Force Office of Scientific Research. REFERENCES 1 J. Cazaux. J. Appl. Phys. 111 (2012) 064903. 2 R. Cimino, I.R. Collins. Applied Surface Science 235 (2004) 231. 3 R. Cimino, I. R. Collins, M. A. Furman, M. Pivi, F. Ruggiero, G. Rumolo, F. Zimmermann, Phys. Rev. Lett. 93 (2004) 014801. 4 R. Cimino. Nuclear Instruments and Methods in Physics Research A 561 (2006) 272. 5 T. Tondu, M. Belhaj, and V. Inguimbert, J. Appl. Phys. 110 (2011) 093301. 6 Y. Raitses, I. D. Kaganovich, A. Khrabrov, D. Sydorenko, N. J. Fisch, and A. Smolyakov. IEEE Transactions on Plasma Science 39 (2011) 995. 7 I. M Bronshtein, B. S Fraiman. Secondary Electron Emission (monograph). Moscow, Russia: Atomizdat (1969). 8 W. Schottky, Phys. Zs. 15 (1914) 872. 9 S. Dushman. Rev. Mod. Phys. 2 (1930) 381. 10 M. P. Seah and W. A. Dench, Surf. Interface Anal. 1 (1979) 2. 11 B. Ziaja, R. A. London, and J. Hajdu, J. Appl. Phys. 99 (2006) 033514. 12 L.D. Landau, E.M. Lifshitz. Quantum Mechanics, Non‐Relativistic Theory: Vol. 3 of Course of Theoretical Physics, JB Sykes, JS Bell, ME Rose - Physics Today, 1958. 13 R. J. Zolweg, Surface Sci. 2 (1964) 409. 14 R. Niedermayer and J. Hölzl, Phys. Stat. Solidi 11 (1965) 651. 15 P. A. Anderson. Phys. Rev. 75 (1949) 1205. 16 J. H. Fritz, C. A. Hague. Rev. Sci. Instrum. 44 (1973) 394. 17 G. A. Haas. J. Vac. Sci. and Technol. 13 (1976) 479. 18 I. M. Bronshtein, V.V. Roshchin, Sov. J. Tech.-Phys. 3 (1958) 2271. 19 I. H. Khan, J. P. Hobson, and R.A. Armstrong, Phys. Rev. 129 (1963) 1513. 20 H. Heil, and J. V. Hollweg, Phys. Rev. 164 (1961) 881. 21 Z. Yakubova and N. A. Gorbatyi, Russian Physics Journal 13 (1970) 1477.", + "(1963) 1513. 20 H. Heil, and J. V. Hollweg, Phys. Rev. 164 (1961) 881. 21 Z. Yakubova and N. A. Gorbatyi, Russian Physics Journal 13 (1970) 1477.", + "Transport signatures of plasmon fluctuations in electron hydrodynamics Dmitry Zverevich and Alex Levchenko Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA (Dated: November 4, 2023) In two-dimensional electron systems, plasmons are gapless and long-lived collective excitations of propa- gating charge density oscillations. We study the fluctuation mechanism of plasmon-assisted transport in the regime of electron hydrodynamics. We consider pristine electron liquids where charge fluctuations are ther- mally induced by viscous stresses and intrinsic currents, while attenuation of plasmons is determined by the Maxwell mechanism of charge relaxation. We show that while the contribution of plasmons to the shear vis- cosity and thermal conductivity of a Fermi liquid is small, plasmon resonances in the bilayer devices enhance the drag resistance. In systems without Galilean invariance, fluctuation-driven contributions to dissipative coefficients can be described only in terms of hydrodynamic quantities: intrinsic conductivity, viscosity, and plasmon dispersion relation. I. INTRODUCTION Hydrodynamic effects in electron transport can occur in solids at intermediate temperatures when the system is suf- ficiently pure, see reviews [1–5] for the detailed discussion of recent results. Indeed, Gurzhi [6] argued early on that if electron-electron interactions provide the most frequent scattering mechanism so that the corresponding mean free path is the shortest length scale in the problem, then one could use approximate local conservation laws to develop an effective hydrodynamic description. The hydrodynamic equations can be obtained by expanding the equations of mo- tion for the element of the fluid in gradients of the velocity and thermodynamic quantities up to terms of second order in the spatial derivatives [7]. Alternatively, these equations can be derived from a more microscopic Boltzmann kinetic the- ory by projecting collision terms into the slow modes [8]. In this principal approximation, the structural form of hydro- dynamic equations follows uniquely from the general con- servation laws of particle, momentum, energy densities, and underlying symmetry of the system. When going beyond the leading approximation one can consider several different types of corrections to hydrody- namics. First are the usual gas-kinetic corrections obtained by Burnett [9] based on Boltzmann equation with an exten- sion of the Chapman-Enskog expansion method. These cor- rections lead to the appearance of the higher-order gradi- ent terms. This approach can be made systematic in dilute systems, e.g. gases, however, it is uncontrolled for fluids where there is no small parameter. Second, are the corre- lation effects that can arise in electron systems subjected to the long-range disorder potential such as strongly-correlated high-mobility semiconductor devices [10, 11] and graphene devices [12, 13]. It can be shown that in these systems hydrodynamic equations retain their principal form on the scales large as compared to the disorder correlation radius but with the renormalized quantities and dissipative coeffi- cients. Third is thermal fluctuation corrections in pristine systems due to the presence of long-lived collective modes, particularly acoustic fluctuations. It was shown by Andreev [14] that the fluctuation mechanism is always the basic one at sufficiently small gradients. In neutral normal fluids long- wavelength thermal fluctuations result in nonanalytical cor- rections thus leading to essentially nonlocal equations.", + "systems due to the presence of long-lived collective modes, particularly acoustic fluctuations. It was shown by Andreev [14] that the fluctuation mechanism is always the basic one at sufficiently small gradients. In neutral normal fluids long- wavelength thermal fluctuations result in nonanalytical cor- rections thus leading to essentially nonlocal equations. In- terestingly, these corrections contain no new parameters and can be expressed solely in terms of the thermodynamic func- tions and dissipative coefficients that enter the hydrody- namic equations in the main approximation. In electron liquids, the relevant collective modes are prop- agating charge density fluctuations — plasmons [15]. In two- dimensional electron systems, plasmons are low-lying gap- less and long-lived excitations [16, 17]. While plasmons were meticulously studied over the years, most recently in the con- text of graphene and surface states of topological insulators [18–34], the role of plasmons in electron hydrodynamic be- havior was not systematically addressed. In the hydrody- namic limit, plasmons can be thermally excited by fluctuat- ing viscous stresses and intrinsic currents. In bilayer devices, one finds both acoustic and optical plasmons branches. In systems without Galilean invariance, these modes are atten- uated by the Maxwell mechanism of charge relaxation. In the Galilean invariant case, when intrinsic conductivity van- ishes, the decay of plasmons is governed by viscous diffu- sion. Our estimations show that fluctuation corrections of propagating plasmons to viscosity and thermal conductivity of a Fermi liquid are small. We further identify examples of transport phenomena where plasmons play a dominant role in the hydrodynamic regime. Building on our recent work on the near-field energy transfer [35] we consider Coulomb drag resistance in the electronic double-layers. The presentation is organized as follows. In Sec. II we sum- marize the main ingredients of the theory of hydrodynamic fluctuations in electron liquids that form the basis of our anal- ysis. This presentation parallels earlier works [36, 37] on the technically overlapping topics. In Sec. III we apply this for- malism to determine the nonequilibrium dynamical structure factor of an electron fluid to the linear order in the hydrody- namic velocity. This result enables the calculation of the drag resistance near plasmon resonances in electron double-layer devices. In Sec. (IV) we introduce the kinetic equation for plasmons in the relaxation time approximation and apply it to estimate the contribution of plasmons to the viscosity and thermal conductivity of electron fluid. As any real device is prone to some degree of disorder in Sec. V we recall the im- pact of impurities and long-range density inhomogeneities on the plasmon broadening. arXiv:2306.13534v2 [cond-mat.mes-hall] 4 Nov 2023 2 II. HYDRODYNAMIC FLUCTUATIONS To have a self-contained presentation we provide a brief account of the hydrodynamic theory with an inclusion of stochastic Langevin forces. For normal fluids this formalism was developed by Landau and Lifshitz [38], and generalized by Khalatnikov [39] for the case of superfluids. The pedagog- ical presentation of lectures on hydrodynamic fluctuations can be found in Refs. [40, 41]. In applications to electron liq- uids in quantum materials one has to add the Coulomb law as appropriate for the charged", + "and Lifshitz [38], and generalized by Khalatnikov [39] for the case of superfluids. The pedagog- ical presentation of lectures on hydrodynamic fluctuations can be found in Refs. [40, 41]. In applications to electron liq- uids in quantum materials one has to add the Coulomb law as appropriate for the charged system, and incorporate ad- ditional terms in transport currents for the systems that are generically not Galilean invariant. For this purpose and having in mind applications to interactively-coupled transport in electron double-layered devices, we consider a planar geometry of two conducting two-dimensional sheets separated by the distance d. Applica- bility of the hydrodynamic approximation requires us to con- sider transport regime when the intralayer electron mean- free path l is short as compared to the interlayer spacing, l ≪d. In each layer the thermally-driven spatial and tem- poral fluctuations of the particle density δn(r, t) render the corresponding current fluctuations δjn(r, t) that follow each other in accordance with the continuity equation [7] ∂tδn + ∇· δjn = 0. (1) Fluctuations of the particle current density comprise of sev- eral contributions [42] δjn = vδn + nδv + σ e2 δF + δξ. (2) The first term in the above expression is the convective part of fluctuations in the presence of macroscopic hydrodynamic flow of the fluid with the velocity v(r, t). The second term describes fluctuations in the hydrodynamic velocity. The third term captures current fluctuations generated by fluc- tuating electromotive force δF . This term is present in sys- tems with broken Galilean invariance, which have nonva- nishing intrinsic conductivity σ. The last term is dictated by the fluctuation-dissipation theorem and describes random Langevin currents whose correlation function at temperature T is given by [43] ⟨δξi(r, t)δξj(r′, t′)⟩= 2T σ e2 δijδ(r −r′)δ(t −t′), (3) where ⟨. . .⟩denotes thermal average [44]. In principle, par- ticle current fluctuations in Eq. (2) may also include thermo- electric contributions generated by the temperature fluctua- tions, e.g. α∇δT, where α is an intrinsic thermoconductiv- ity. However, for the type of effects that we consider below, these terms lead to insignificant corrections, which we thus neglect. Thermal fluctuations also lead to the entropy density fluc- tuations of the fluid δs(r, t) and associated with it fluctu- ations of the respective entropy current density δjs(r, t). These quantities are also linked by the continuity equation ∂tδs + ∇· δjs = 0. (4) Without thermoelectric effects the entropy current density also has four contributions similar to Eq. (2) δjs = vδs + sδv −κ T ∇δT + δζ T . (5) Here the first two terms are completely analogous to that in Eq. (2). The third term describes entropy fluxes due to fi- nite thermal conductivity κ. The last term is the associated Langevin thermal noise whose correlation function is given by ⟨δζi(r, t)δζj(r′, t′)⟩= 2κT 2δijδ(r −r′)δ(t −t′). (6) For a fluid with the mass density ρ the evolution of the momentum density p = ρv is governed by the Navier-Stokes equation ρ(∂t + v · ∇)δv = −∇δ ˆΠ −en∇δΦ, (7) which we present", + "whose correlation function is given by ⟨δζi(r, t)δζj(r′, t′)⟩= 2κT 2δijδ(r −r′)δ(t −t′). (6) For a fluid with the mass density ρ the evolution of the momentum density p = ρv is governed by the Navier-Stokes equation ρ(∂t + v · ∇)δv = −∇δ ˆΠ −en∇δΦ, (7) which we present here in its linearized form with respect to fluctuations. The first term on the right-hand-side of Eq. (7) captures fluctuations of the momentum flux tensor δΠij = δPδij −δΣij. (8) It includes local hydrodynamics fluctuations in the pressure of a fluid δP = \u0012∂P ∂n \u0013 s δn + \u0012∂P ∂n \u0013 v δs, (9) that couples Eq. (7) to the continuity equations (1) and (4), and fluctuations of viscous stresses δΣij = η(∂iδvj + ∂jδvi −δij∂kδvk) + δΞij, (10) that are expressed via gradients of the velocity field, where η is the shear viscosity. The random viscous Langevin sources in Eq. (10) are described by the correlation function of the form [38] ⟨δΞik(r, t)δΞlm(r′, t′) = 2ηT(δilδkm + δimδkl −δikδlm)δ(r −r′)δ(t −t′). (11) For simplicity, we neglected terms with the bulk viscosity. As is known, bulk viscosity vanishes in the systems with quadratic and linear dispersion relations [8]. Finally, the sec- ond term on the right-hand-side of Eq. (7) describes the flow of momentum due to fluctuations of the long-range Coulomb interaction. The corresponding fluctuations of the electric potential δΦ are related to the electron density fluctuation δn by the Poisson equation. It should be noted that in the ge- ometry of a bilayer δΦ includes both potential due to density fluctuations in a given layer, as well as dynamically screened potential arising from the density fluctuations in the other layer. The formalism of hydrodynamic theory of fluctuations al- lows computation of various correlation functions in any concrete setup. The approach to a given problem of interest is rather straightforward and can be summarized as follows. 3 One considers Langevin sources δξ, δζ, δΞij as given func- tions fluctuating in space and time. The linearized equations of motion then can be solved for δn, δv, δP with an account of the proper boundary conditions. As a result these quanti- ties are exprressed as linear functionals of the source fields. Therefore, any quadratic form with respect to δn, δv, δP can be expressed via quadratic average of sources with the help of the fluctuation-dissipation relations. Upon thermal aver- aging ⟨. . .⟩the auxiliary sources drop out and the result is ex- pressed via a handful number of dissipative coefficients and thermodynamic quantities of the system. In this work we calculate the dynamic structure factor of the fluid which is formally defined as the density-density cor- relation function. This object is finite even in equilibrium, it carries information about the collective modes in the system and obeys rather generic properties such as Kramer-Kronig relations and sum rules [40, 45]. Furthermore, being inter- ested in the transport effects we take one step further, and calculate correction to the structure factor to the linear order in hydrodynamic velocity. These results enable applications to nonlocal transport", + "the system and obeys rather generic properties such as Kramer-Kronig relations and sum rules [40, 45]. Furthermore, being inter- ested in the transport effects we take one step further, and calculate correction to the structure factor to the linear order in hydrodynamic velocity. These results enable applications to nonlocal transport effects in bilayers such as drag friction. III. PLASMON-ENHANCED COULOMB DRAG RESISTIVITY Coulomb drag [46] is the useful experimental technique to directly probe the strength of electronic correlations that can be quantified via measured drag resistance. The setup consists of two spatially-separated and electrically-isolated conducting layers, where one (active) layer is driven out of equilibrium and the resulting nonlocal response is measured in the other (passive) layer that can be dragged along since electrons interact via the long-range Coulomb potential. Im- portantly for our applications, Coulomb drag was recently measured in both monolayer and bilayer graphene double- layers [47–51]. To the best of our knowledge, the existing calculations of this effect [52–59] didn’t address the role of plasmons in the hydrordynamic regime. Thus far such anal- ysis was carried out only in the Galilean invariant systems [36, 60]. Here we provide solution to this problem with gen- eralizations as appropriate for systems with broken Galilean invariance. A. Linear response analysis For simplicity we consider symmetrical layers with aver- age carrier density n. We work in the limit kF d ≫1, where kF is the Fermi momentum. We denote density fluctuations in each layer as δn1,2(r, t) and respective fluctuations of ve- locity as δv1,2(r, t). In the active layer we impose finite (in average) homogeneous hydrodynamic velocity v. For all fluctuating quantities, including Langevin sources, we intro- duce Fourier components {δn, δv, δΦ} ∝exp(−iωt + iqr). In these notations the linearized continuity equation (1) takes the form −iωδn1 + in(q · δv1) + i(q · v)δn1 + σ e2 q2eδΦ1 + i(q · δξ1) = 0. (12) In the fluctuating electromotive force, Coulomb potential in- cludes both fluctuations of the density in the layer-1 as well as screened fluctuations of the density in the layer-2, namely δΦ1(q, ω) = 2πe ϵq \u0002 δn1(q, ω) + e−qdδn2(q, ω) \u0003 , (13) where ϵ is the dielectric constant of the material surrounding the electron layers. The continuity equation for the density fluctuations δn2 in the passive layer is the same, one only has to interchange indices 1 ↔2 and take v →0. The linearized Navier-Stokes equation (7) in the active layer takes the form −iρωδv1 + iρ(q · v)δv1 = −ienqδΦ1 + i(q · δΣ1). (14) Here we made one approximation by neglecting pressure fluctuations δP as compared to fluctuations in the Coulomb potential δΦ. This is legitimate as the long-range nature of Coulomb interaction dominates for fluctuations in the long wave length limit when q →0. We also note that with this approximation, the entropy continuity equation decou- ples and thus entropy fluctuations will not contribute the to drag. Previous analysis [36] showed that density-density coupling induced by thermal expansion of the fluid and tem- perature fluctuations leads to a very small drag", + "when q →0. We also note that with this approximation, the entropy continuity equation decou- ples and thus entropy fluctuations will not contribute the to drag. Previous analysis [36] showed that density-density coupling induced by thermal expansion of the fluid and tem- perature fluctuations leads to a very small drag that can be disregarded. It is convenient to multiply Eq. (14) by one ex- tra power of q to have equation in the scalar form. We then notice that q · (q · δΣ) = iηq2(q · δv) + q · (q · δΞ). As a next step, we rewrite Eqs. (12) and (14) equivalently n(q · δv1) −ωδn1 −iγq(δn1 + e−qdδn2) = −(q · v)δn1 −(q · δξ1), (15a) (ω + iωη)n(q · δv1) −ω2 p(δn1 + e−qdδn2) = n(q · v)(q · δv1) −n ρ q · (q · δΞ1), (15b) where we introduced ωq = s 2π(ne)2q ρϵ , γq = 2πσq ϵ , ωη = ηq2 ρ . (16) There are two more equations of the same kind for the passive layer. At this point, one can exclude fluctuations of velocity to arrive at the coupled equations that govern density fluctu- ations only. For this purpose, it is convenient to symmetrize density fluctuations and introduce δn± = δn1 ± δn2, and similarly for all otgher quantities e.g. δξ± = δξ1 ± δξ2. In this basis of normal modes fluctuations δn± decouple. Fi- nally, we introduce δn± = δn(0) ± + δn(1) ± + . . ., where δn(0) ± denote the equilibrium fluctuations induced by the Langevin sources, and δn(1) ± capture the nonequilibrium advection of 4 fluctuations by the hydrodynamic flow to the linear order in v. After some algebra we find δn(0) ± = −n ρΓ± q · (q · δΞ±) + ω + iωη Γ± (q · δξ±), (17a) δn(1) ± = (q · v) 2Γ± X ± h Υ±δn(0) ± −(q · δξ±) i . (17b) Here we introduced Γ±(q, ω) = ω2 −ω2 ± −ωηγ± + iω(ωη + γ±), (18a) Υ±(q, ω) = 2ω + i(ωη + γ±), (18b) where ω2 ± = ω2 q(1 ± e−qd), γ± = γq(1 ± e−qd). (19) To close the system of equations we need thermal averages which follow from Eqs. (3) and (11) ⟨(q · δξ±)(q · δξ±)⟩= 4Tq2σ/e2, (20a) ⟨q · (q · δΞ±)q · (q · δΞ±)⟩= 4Tq4η. (20b) It is clear from the obtained solution that the dynamical struc- ture factor ⟨δn±δn±⟩contains resonant denominators for certain values of frequencies and wave numbers of fluctua- tions. We thus first discuss roots of Γ±(q, ω). B. Hydrodynamic plasmon modes As is well known from the hydrodynamic linear response theory [40], zeros of Γ±(q, ω) define complex frequencies of collective modes propagating in the system. The real part of frequency defines the dispersion relation of the mode, while the complex part defines its decay rate. The double-layer sys- tem supports two modes termed as optical plasmon (OP) and acoustic plasmon (AP) [16, 17]. This distinction holds for qd ≪1, while in the opposite limit dispersions smoothly", + "real part of frequency defines the dispersion relation of the mode, while the complex part defines its decay rate. The double-layer sys- tem supports two modes termed as optical plasmon (OP) and acoustic plasmon (AP) [16, 17]. This distinction holds for qd ≪1, while in the opposite limit dispersions smoothly connect to the plasmon frequencies of each individual lay- ers. The OP corresponds to the in-phase oscillations of the electron density and its dispersion relation is similar to the plasmon frequency of a single layer with the square-root de- pendence on q, ω+ ∝√q. The AP mode corresponds to the out-of-phase charge neutral oscillations and thus exhibits lin- ear dispersion ω−∝q. Attenuation of plasmons occurs differently in systems with or without Galilean invariance. Indeed, in the Galilean in- variant system, σ →0, the imaginary part of the root of Γ± depends only on ωη. Therefore, relaxation occurs via vis- cous diffusion and the corresponding rate is the same for both OP and AP that scales quadratically with the wave vector, namely ℑω ∝q2. The fact that kinematic viscosity deter- mines plasmon decay was discussed earlier in the context of conductivity of the classical two-dimensional electron gas [61]. If Galilean invariance is broken, the relaxation is domi- nated by the Maxwell mechanism of charge dissipation. This feature is important and was discussed in the context of non- local optical conductivity in graphene [62]. In a double-layer, the decay rate of OP is linear in q whereas it scales quadrat- ically for the AP. We see that regardless of the attenuation mechanism, both branches of charge oscillations are under- damped so that plasmons are well-defined and long-lived ex- citations in the hydrodynamic regime. We note that plas- mons remain underdamped in the high-frequency (kinetic) regime, ω ≫vF q, where plasmon attenuation is dominated by the decay into two part particle-hole pairs. From the re- sults of Refs. [63, 64] we can deduce that the corresponding rates are the same for both Galilean invariant and Dirac sys- tems, modulo logarithmic factors, and scale as ℑω ∝q2. Dis- order leads to additional mechanism of plasmon broadening that we discuss separately in Sec. IV. C. Drag force and resistivity We apply the above formalism to the Coulomb drag prob- lem. The steady current in the active layer exerts the drag force on the passive layer. We relate the potential to den- sity fluctuations by using the Poisson equation and thus ex- press the drag force in terms of the density-density correla- tion function FD = Z dωd2q (2π)3 (−iq)2πe2 ϵq e−qd⟨δn1(q, ω)δn2(−q, −ω)⟩. (21) Knowing the drag force one readily finds the drag resistivity rD = FD/(e2n2v). (22) We observe that since viscous stresses do not correlate with intrinsic current fluctuations the respective contributions to the drag force can be considered separately. We focus on the latter terms as they yield the dominant contribution. To calculate the average ⟨δn1δn2⟩we transform it into the symmetrized basis. We notice that equilibrium parts of the average ⟨δn(0) ± δn(0) ± ⟩do not contribute to the force as they are isotropic", + "force can be considered separately. We focus on the latter terms as they yield the dominant contribution. To calculate the average ⟨δn1δn2⟩we transform it into the symmetrized basis. We notice that equilibrium parts of the average ⟨δn(0) ± δn(0) ± ⟩do not contribute to the force as they are isotropic thus average to zero upon q integration. To the linear order in v we have two types of terms to con- sider: the same parity ⟨δn(0) ± δn(1) ± ⟩and the mixed parity ⟨δn(0) ± δn(1) ∓⟩contributions. Using Eq. (17) we notice that both Γ± and Υ± depend only on the absolute value of the wave vector q = |q|, therefore they are even function with respect to q →−q exchange. In contrast, changing the sign of frequency, ω →−ω we get Γ±(q, −ω) = Γ∗ ±(q, ω) and Υ±(q, −ω) = −Υ∗ ±(q, ω). These properties result in the correlator ⟨δn(0) ± δn(1) ± ⟩being frequency odd while correlator ⟨δn(0) ± δn(1) ∓⟩being frequency even. Thus, the former drops out upon the frequency integration in Eq. (21). The remain- ing contributions come in the complex conjugated pairs. To simplify the formulas at the intermediate steps we notice that ω2 +γ−−ω2 −γ+ = 0, which gives ℑ(Π+Π∗ −) ≈ω3(γ+ −γ−) and ℑ(Υ∗ −Γ+ + Υ+Γ∗ −) ≈3ω3(γ+ −γ−). The approximate 5 FIG. 1. Dimensionless function F(x, y, β) defined by Eq. (25) plot- ted for β = 0.15. The traces of plasmon resonances can be seen for momenta x ≳1 as their spectral weight is suppressed at x →0 thus making them invisible. The dimensionless parameter β controls the broadening of plasmon branches and determines the maximum of F. sign means that we neglected the shift of the plasmon pole ∝ωηγ± in the expressions for Γ±, and we also neglected viscous damping of plasmons as compared to γ± in the imag- inary part of Γ± and Υ±. Collecting all the pieces together we arrive at the following expression rD = Tσ 2e4n2 Z dωd2q (2π)3 \u00122πe2 ϵq \u0013 ω4q4(γ+ −γ−)e−qd |Γ+|2|Γ−|2 . (23) To deal with final integrations it is useful to rescale all fre- quencies ω in units of the plasma frequency ωq taken at q = 1/d, and also to rescale all momenta q in units of 1/d. The integral then depends on a single dimensionless param- eter β = (γq/ωq)q=1/d. These steps lead us to the final result for the drag resistance rD = σ 4π2e4 \u0012 T EF \u0013 \u0012 1 nd2 \u00132 f(β). (24) The dimensionless function f(β) is obtained from the double- integral of the following function F(x, y, β) = Y ± √βx5/2y2e−x (y2 −x(1 ± e−x))2 + β2y2x2(1 ± e−x)2 (25) where x = qd and y = ω/ωq=1/d. In order to highlight importance of plasmons we plot F(x, y, β) in Fig. (1). Physi- cally this function is defined by the product of the dynamical structure factor, the phase space volume, and the strength of Langevin fluxes. We see that the resonant contribution of plasmons to the drag resistance occurs for q ∼1/d", + "of plasmons we plot F(x, y, β) in Fig. (1). Physi- cally this function is defined by the product of the dynamical structure factor, the phase space volume, and the strength of Langevin fluxes. We see that the resonant contribution of plasmons to the drag resistance occurs for q ∼1/d and ω ∼ωq=1/d, or equivalently in dimensionless notations x ∼y ∼1. For (x, y) ≪1 the spectral weight of plasmon resonances is suppressed and they are not clearly visible on the plot. On Fig. 2 we further plot dispersions of optical and acoustic plasmons superimposed on top of the color plot of the function F(x, y, β). We estimate the dimensionless pa- FIG. 2. Dispersion laws ω = ω±(q) for OP and AP shown by dashed lines superimposed on top of the color plot that defines magnitude of the dimensionless function from Fig. 1 plotted for β = 0.15. The light area where OP and AP tend to merge corresponds to the maximum of F ∼4. rameter β to be of the order β ∼σ e2 s e2 ϵvF 1 √kF d. (26) Since we work in the limit kF d ≫1 it is small. In the limit β ≪1 we are able to extract an asymptotic expression for f(β) in the form f(β) ≈π 10 ln3(Λ) ln 4 β ln Λ β ! , Λ = 2 β q ln 1 β . (27) To assess the accuracy of this approximate formula we com- pare it to the result of the numerical integration and find an excellent agreement, see Fig. 3. In a similar way one can compute viscous contribution to the drag resistivity from Eqs. (17) and (21). The final result can be found in the form rD = ϵvF e4 \u0012 T EF \u0013 η/n (kF d)5 g(β), (28) where g(β) is another dimensionless function that has log- arithmic dependance on β in the limit β ≪1. There are two key differences between rD given by Eqs. (24) and (28). First is that intrinsic mechanism is parametrically stronger as it decays as 1/d4 whereas viscous contribution diminishes much faster as 1/d5 at large interlayer separations. There- fore, drag effect is stronger in systems with broken Galilean 6 0.05 0.10 0.15 0.20 0.25β 10 20 30 40 50 f(β) FIG. 3. Plot of the dimensionless function f(β) introduced in Eq. (24). The solid line represents the result of a numerical integration, and the dashed line corresponds to the approximate analytical for- mula given in Eq. (27) and applicable for β < 1. invariance. This feature is qualitatively consistent with ex- perimental observations. The second difference concerns the temperature dependence. In the Fermi liquid regime η ∝ 1/T 2 therefore viscous contribution scales as rD ∝1/T in the hydrodynamic limit. In contrast, for systems with bro- ken Galilean invariance the temperature dependence of rD is determined by the product of intrinsic conductivity σ(T) and an extra power of T coming from the Langevin sources, modulo additional logarithmic term in f(β). For example, in the monolayer and bilayer graphene, σ(T) is known", + "contrast, for systems with bro- ken Galilean invariance the temperature dependence of rD is determined by the product of intrinsic conductivity σ(T) and an extra power of T coming from the Langevin sources, modulo additional logarithmic term in f(β). For example, in the monolayer and bilayer graphene, σ(T) is known to be very weakly temperature dependent, therefore rD is approx- imately linear in temperature, rD ∝T. It is striking to ob- serve that at temperatures where l ∼d the hydrodynamic re- sult for drag resistance is parametrically larger than the con- ventional Fermi liquid result for the collisionless regime [65– 67]. This implies that frequent intralayer collisions strongly enhance Coulomb drag. We note that Eq. (28) applies to the Galilean invariant systems except that β should be replaced by a different dimensionless parameter that depends on vis- cosity β →η/(ρd2ωq) with q = 1/d [36]. Perhaps even more importantly, Eqs. (24) and (28) apply to non-Fermi liquids as long as hydrodynamic limit can be justified [68]. IV. PLASMON CORRECTIONS TO KINETIC COEFFICIENTS IN FERMI LIQUIDS As mentioned in the introduction, the fluctuational cor- rections to the viscosity of a two-dimensional neutral liquid diverge [14], which means that hydrodynamic equations are nonlocal in two dimensions. On the other hand, the calcula- tions presented in the preceding sections were based on the local form of the theory and assumed that the viscosity and thermal conductivity of the electron system are well-defined quantities. Thus it is instructive to estimate fluctuational corrections to the dissipative hydrodynamic coefficients of the two-dimensional charged electron liquid. We present a straightforward generalization of the procedure outlined in Ref. [61] to the case of systems with broken Galilean invari- ance. For this purpose it is convenient to introduce a distri- bution function of plasmon fluctuations Nq that obeys the kinetic equation ∂Nq ∂t + ∂ωq ∂q ∇Nq = St{Nq}. (29) The microscopic derivation of the collision integral St{Nq} that describes scattering of plasmons is a challenging task [69]. For the sake of estimation it suffice to use the relaxation-time-approximation for the linearized collision in- tegral, St{Nq} = −γqδNq, where δNq denotes the nonequi- librium part of the distribution and the intrinsic relaxation rate is given by Eq. (16). The energy flux of plasmons can be written in the usual kinetic form jε = Z d2q (2π)2 vqωqNq, (30) where vq = ∂ωq/∂q is the group velocity of plasmons. Solv- ing the kinetic equation to the linear order in a small tem- perature gradient ∇T gives correction to the thermal con- ductivity jε = −δκ∇T, δκ = 1 2 Z ωqτqv2 q ∂Nq ∂T d2q (2π)2 , (31) where τq = 1/γq and Nq should be understood as the equi- librium Bose distribution function. In two-dimensional pure systems the momentum integral logarithmically diverges at the infrared. However, in the presence of the elastically scat- tering potential the dispersion relation of plasmons which we used holds only at those frequencies for which the plasmon mean free time is shorter than the electron-impurity scatter- ing time τ. This determines a cut-off in Eq. (31).", + "diverges at the infrared. However, in the presence of the elastically scat- tering potential the dispersion relation of plasmons which we used holds only at those frequencies for which the plasmon mean free time is shorter than the electron-impurity scatter- ing time τ. This determines a cut-off in Eq. (31). In Sec. V we analyze broadening of plasmon dispersion and provide esti- mates for two different models of the disorder potential. As a result, the fluctuational correction to the thermal conductiv- ity can be estimated as δκ ∼(e2/σ)EF ln(Tτ). This estimate should be compared to the thermal conductivity of a two- dimensional electron gas with Coulomb interaction [70, 71]: κ = (E2 F /T) ln−1(EF /T). Therefore the correction is small δκ/κ ∼(T/EF ) ≪1. In complete analogy we can estimate corrections to vis- cosity. For that one needs to consider the flux of momentum associated with plasmon excitations, which is given by Πij = − Z d2q (2π)2 qi ∂ωq ∂qj Nq. (32) Solving now the kinetic equation (29) to the linear order in the hydrodynamic flow with velocity u(r) we extract the correction to the shear viscosity δη = −1 2 Z q2 \u0012∂ωq ∂q \u00132 τq ∂Nq ∂ωq d2q (2π)2 , (33) and after integration δη ∼(v2 F /e2σ)n(T/EF )3. This correc- tion is negligible as compared to the electron viscosity in a 2D Fermi liquid, which is η ∼n(EF /T)2 modulo logarith- mic factors [72, 73]. 7 V. PLASMON ATTENUATION FROM DISORDER SCATTERING For completeness, we briefly recall the effect of disorder on plasmon scattering and relaxation, see Refs. [23, 24, 33, 62] for the related studies. To this end, it suffices to analyze the σ →0 limit, namely Galilean invariant case. In the follow- ing, we focus on a specific experimentally relevant setup in which a doping layer is separated from the two-dimensional electron system by a distance d [17]. Randomness of the spa- tial distribution of charged dopants leads to electron density in the form n(r, t) = n + en(r) + δn(r, t) (34) where n is the average uniform electron density, en(r) is the static density variations due to doping layer, and δn(r, t) is the plasmon-related dynamic density oscillations. The av- erage density of dopants is equal to the average density of electrons. Within the hydrodynamic approximation density fluctuations follow the equation of motion ∂2 ∂t2 δn(r, t)−e2 ϵm∇[n+en(r)]∇ Z d2r′ δn(r′, t) |r −r′| = 0 (35) In the Fourier representation this equation can be cast in the form (ω2 −ω2 q)δn(q) = 2πe2q ϵm X k (nq · nk)en(q −k)δn(k) (36) where nk is the unit vector along the direction of the corre- sponding momentum. We solve this integral equation pertur- batively performing disorder averaging to the lowest order in en. After one iteration we find ω2 −ω2 q −Σ(ω, q) = 0, (37) where the self-energy is given by Σ(ω, q) = \u00122πe2 ϵm \u00132 X k qk(nq · nk)2 ω2 −ω2 k + i0D(q −k), (38) which is expressed in terms of a correlation function of den- sity variations", + "After one iteration we find ω2 −ω2 q −Σ(ω, q) = 0, (37) where the self-energy is given by Σ(ω, q) = \u00122πe2 ϵm \u00132 X k qk(nq · nk)2 ω2 −ω2 k + i0D(q −k), (38) which is expressed in terms of a correlation function of den- sity variations caused by disorder D(q) = Z d2r⟨en(r)en(0)⟩e−iqr. (39) In contrast to the notations of the previous sections here ⟨. . .⟩denotes disorder average rather than thermal average. It should be noted that within the theory of linear screening this correlation function is related to the probability of elec- tron scattering since density variations cause potential varia- tions, νδV (r)+en(r) = 0, which is Thomas-Fermi condition, and ν = m/π is the density of states. The imaginary part of the self-energy in Eq. (38) yields the scattering rate τ −1 q = −1 2ωq ℑΣ(ωq, q) = π2e2q 2ϵmn X k (nq · nk)2δ(ωq −ωk)D(q −k) (40) For the spatially uncorrelated dopants, the spectral power of the external random potential induced in the plane of the electron system is D(q) = nκ2 (q + κ)2 e−2qd, (41) where κ = 2πe2ν is the inverse Thomas-Fermi screening radius. For a quantum well of a thickness a the above ex- pression for D(q) should be additionally multiplied by a fac- tor sinh(qa)/qa [17]. In Eq. (41) we assumed qa ≪1. For smooth disorder (kF , κ) ≪1/d, which leads to a plasmon broadening τ −1 q = ωqq2 16πn Z π 0 dθ cos2 θe−4qd sin θ 2 = qωq 32πnd \u001a πqd qd ≪1 1 qd ≫1 . (42) It can be readily verified that the effect of density varia- tion in the random long-range potential on the attenuation of two-dimensional plasmons dominates over the effect of broadening of single-particle states for smooth disorder. In- deed, the latter can be inferred from the optical Drude con- ductivity σ(ω) = ie2n/m ω + i/τtr (43) expressed in terms of the transport scattering time [74] τ −1 tr = 2π ν Z dn′(1 −n · n′)D(kF n −kF n′). (44) In the hydrodynamic approach [16] the plasmon spec- trum is related to the optical conductivity as follows ω = −2πiqσ(ω)/ϵ, which is equivalent to ω(ω + i/τtr) = ω2 q. (45) This equation is analogous to Eq. (37). However in contrast, plasmon broadening is independent of q in the limit τ −1 tr ≪ ωq, and simply determined by the transport scattering rate τ −1 q = 1 2τtr = 2πn ν Z π 0 dθ sin2 θ 2e−4kF d sin θ 2 ≈ EF (2kF d)3 . (46) It is dominated by a small angle scattering θ ∼q/kF ≪ 1. Comparing Eq. (42) and (46) one can see that plasmon broadening from inhomogeneous density variation domi- nates starting from momenta q ∼d−1/(kF d)1/5 < d−1. In contrast, if we compared now intrinsic Maxwellian rate of attenuation for systems with broken Galilean invariance Eq. (16) to that of disorder broadening Eq. (42) at typical moment for bilayers q ∼d−1 the former dominates in a large factor", + "domi- nates starting from momenta q ∼d−1/(kF d)1/5 < d−1. In contrast, if we compared now intrinsic Maxwellian rate of attenuation for systems with broken Galilean invariance Eq. (16) to that of disorder broadening Eq. (42) at typical moment for bilayers q ∼d−1 the former dominates in a large factor ∼(σ/vF )(kF d)3/2 ≫1. These estimates justify assump- tions and approximation used in Sec. III. 8 ACKNOWLEDGMENTS We thank A. Andreev, L. Fritz, L. Glazman, and B. Spi- vak for discussions that shaped understanding of physics presented in this work. This research project was finan- cially supported by the National Science Foundation Grant No. DMR-2203411 and H. I. Romnes Faculty Fellowship pro- vided by the University of Wisconsin-Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation. 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B 32, 8442–8444 (1985).", + "arXiv:1610.03221v1 [cond-mat.str-el] 11 Oct 2016 2D surprises at the surface of 3D materials: confined electron systems in transition metal oxides Emmanouil Frantzeskakis,1, ∗Tobias Chris R¨odel,1,2 Franck Fortuna,1 and Andr´es Felipe Santander-Syro1, † 1CSNSM, Universit´e Paris-Sud, CNRS/IN2P3, Universit´e Paris-Saclay, 91405 Orsay cedex, France 2Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin - BP 48, 91192 Gif-sur-Yvette, France The scope of this article is to review the state-of-the-art in the field of confined electron systems generated at the bare surfaces of transition metal oxides (TMOs). This scientific field is a prime example of a domain where two-dimensional physics and photoemission-based spectroscopic techniques have together set up the de- velopment of the story. The discovery of a high-mobility two-dimensional electron system (2DES) at interfaces of transition metal oxides has attracted an immense scientific interest due to new opportunities opened in the emerging field of oxide electronics. The subsequent paradigm shift from interfaces to the bare surfaces of TMOs made the confined electron system accessible to surface-sensitive spectroscopic techniques and this new era is the focus of the present article. We describe how results by means of Angle-Resolved Photoemission Spec- troscopy (ARPES) establish the presence of confined electron carriers at the bare surface of SrTiO3(100), which exhibit complex physics phenomena such as orbital ordering, electron-phonon interactions and spin splitting. The key element behind the 2DES generation is oxygen vacancies. Moreover, we review the experimental evi- dence on the generation of 2DESs on surfaces with different orientation, as well as on different TMO substrates. The electronic structure of the confined electron system responds to such changes, thereby providing external means for engineering its properties. Finally, we identify new directions for future research by introducing a device-friendly fabrication protocol for the generation of 2DESs on TMO surfaces. I. Introduction In scientific literature, the term “functional oxides” tradi- tionally refers to the family of transition metal oxides (TMOs) composed of oxygen anions and metals with partially-filled d- shells. TMOs are of high technological importance as cata- lysts [1], superconductors [2, 3], novel platforms for electron- ics [4] and building blocks for spintronic devices [5]. More- over, in terms of accessing novel physics phenomena, the interplay of d electrons in transition metal oxides offers ac- cess to diverse quantum ground states including metallic and semiconducting phases, (anti-)ferromagnetism, ferroelectric- ity, multiferroic behavior, superconductivity, metal-insulator transitions, charge ordering and orbital ordering [6]. The interest of the scientific community on transition metal oxides was rekindled in 2004 by the intriguing possibility to exploit certain oxide interfaces as platforms for oxide electronics [4, 7]. The discovery of a high mobility two- dimensional electron system (2DES) on the interface between two otherwise insulating oxides, namely LaAlO3 (LAO) and SrTiO3 (STO) [8], drove the subsequent scientific effort to- wards exploiting the unexpected 2DES and understanding its emergence [9–13]. The 2DES at the LAO/STO interface is very important not only because it gives rise to 2D conductiv- ity but also because it is at the origin of magnetism, supercon- ductivity and their coexistence, all of them controllable by a gate-voltage [14–17]. This system is therefore a vivid illus- tration", + "The 2DES at the LAO/STO interface is very important not only because it gives rise to 2D conductiv- ity but also because it is at the origin of magnetism, supercon- ductivity and their coexistence, all of them controllable by a gate-voltage [14–17]. This system is therefore a vivid illus- tration of exotic physics phenomena in low dimensions and has a high potential for technological applications. In short, LAO/STO has been considered as the prime candidate for ox- ide electronics not only because it is the most well-studied ox- ide system which hosts a high-mobility 2DES, but also due to the scarcity of alternative interfaces [18–20]. Its high poten- ∗emmanouil.frantzeskakis@csnsm.in2p3.fr †andres.santander@csnsm.in2p3.fr tial led to the term “functional oxides” being used primarily for variations of the LAO/STO interface. A second set of breakthrough studies came in early 2011 and showed that the LAO/STO interface is not necessary to obtain a high mobility 2DES: a similar 2DES was discovered at the bare (100) surface of SrTiO3 [21, 22]. These results opened a whole new line of research. The 2DES became accessible to surface sensitive techniques as it was no longer buried under layers of LaAlO3 but it could be probed right on the sample surface -and subsurface layers. This is in contrast to interface systems where access to the confined electronic structure is very difficult and with poor resolution [23–25]. For 2DESs at TMO surfaces, Angle Resolved Photoemission Spectroscopy (ARPES) became the leading experimental technique in this field due to its high surface sensitivity and the possibility for a direct and high-resolution view of the electronic structure of the 2DES. In the present work we will show how this spectroscopic technique resulted in a deep understanding of the confined electron systems on the TMO surfaces and paved the way for further research. Our goal is to present a combined overview of ARPES studies in this field and bring into focus the latest advances made by our research team at CSNSM. We will start by discussing the characteristics of the 2DES on SrTiO3(100): details of its electronic structure can lead to an understanding of the 2DES origin and reveal the effect of more advanced concepts such as many-body interactions. After reviewing the basic and more advanced characteristics of the 2DES on SrTiO3(100), we will discuss 2DESs on other TMO surfaces: different surfaces of STO give the possibility of orientational tuning of the 2DES, KTaO3 reveals the effect of bulk spin-orbit coupling, TiO2 anatase is an example of a non-perovskite TMO and BaTiO3 shows that a ferroelectric compound can also host a 2DES. Finally, we will finish by bringing into focus a new versatile technique with a high potential of generating device-friendly TMO/2DES systems. 2 II. Experimental Details This section is a summary of the experimental details on sample preparation and on ARPES measurements performed by our team at CSNSM. Further details are explained in Ref. 26 or in the references mentioned in the figure captions. Transition metal oxides went through various steps of ex- situ and in-situ treatment. Before introducing them into the UHV setup, SrTiO3", + "on sample preparation and on ARPES measurements performed by our team at CSNSM. Further details are explained in Ref. 26 or in the references mentioned in the figure captions. Transition metal oxides went through various steps of ex- situ and in-situ treatment. Before introducing them into the UHV setup, SrTiO3 single crystals were etched in buffered HF to create large TiO2-terminated terraces and they were subse- quently annealed under O2 flow to heal the etch holes with a good control of bulk oxygen vacancies. In order to obtain a clean and well-ordered surface, in-situ treatment of the dif- ferent TMO surfaces requires either fracturing or a combina- tion of ion bombardment and/or annealing in UHV conditions. The detailed protocol of in-situ preparation varies for different TMOs. SrTiO3 was prepared either by fracturing or anneal- ing. For the latter we used temperatures of the order of 600oC for 1-2 h as higher temperatures or longer annealing times are known to promote surface reconstructions [27–29]. Clean sur- faces of KTaO3 and TiO2 anatase were exclusively produced by fracturing in UHV conditions. In all cases, the cleanli- ness and order of the surfaces were checked by Auger Elec- tron Spectroscopy (AES) and Low Energy Electron Diffrac- tion (LEED). Monodomain BaTiO3 thin films were grown along the <001> direction by Pulsed Laser Deposition (PLD) using a Kr-F excimer laser. The growth of the film was moni- tored by RHEED. Prior to ARPES experiments the films were annealed at 550oC to remove contamination produced during sample transfer. After obtaining a clean surface, additional capping with pure aluminum was used in the cases of SrTiO3, anatase and BaTiO3 in order to generate oxygen vacancies through a re- dox reaction (see part E of the next section). Aluminum was depositied by means of a Knudsen cell with an alumina cru- cible. The approximate growth rate was 0.3 ˚A/min and the flux was calibrated by means of a quartz microbalance. In or- der to create a high concentration of oxygen vacancies on the near-surface region, amorphous Al films with a thickness of 2 ˚A were deposited on the clean TMO surfaces. ARPES experiments were performed at the Synchrotron SOLEIL (France) and at the Synchrotron Radiation Center (SRC, University of Wisconsin, Madison) using Scienta R4000 hemispherical analyzers with vertical slits. We used variable photon energy, typically in the range of 20-120 eV both with linear horizontal and linear vertical polarization. Measurements were performed at 8K (SOLEIL) or at 25K (SRC) without observing any major T-dependence between these two temperature values. In all cases the pressure was in the range of 10−11 mbar. The typical angle and energy resolutions were 0.25o and 15 meV, respectively, while the mean diameter of the incident photon beam was of the order of 50 µm (SOLEIL) and 150 µm (SRC). III. Results and Discussion A. 2DES on a bare surface of SrTiO3: discovery and basic characteristics An experimental ARPES work by the CSNSM team and collaborators [21], along with a contemporary study by Meevasana et al. [22] were the first studies to report a 2DES on", + "and 150 µm (SRC). III. Results and Discussion A. 2DES on a bare surface of SrTiO3: discovery and basic characteristics An experimental ARPES work by the CSNSM team and collaborators [21], along with a contemporary study by Meevasana et al. [22] were the first studies to report a 2DES on a bare TMO surface setting the scientific background for further work. The key idea behind these studies is that an elec- tric field at the sample surface lifts the degeneracy of the t2g states and confines them into a series of subbands by gener- ating a depth-dependent potential that acts as a quantum well [Figs. 1(a)-1(d)]. Santander et al. approximated the confin- ing potential by a potential wedge Vconf = (z −d)F, where F is the strength of the electric field, z = 0 corresponds to the sample surface and z = d corresponds to the maximum depth of Vconf(z) [21]. Under this approximation, the solution of the 1D Hamiltonian for non-interacting electrons trapped in this potential well yields the quantized eigenenergies of the confined subbands [21]: En = Vconf(0) + \u0012 ℏ2 2m∗z \u0013−1 3 \u0014\u00123π 2 \u0013 \u0012 n −1 4 \u0013 eF \u0015 2 3 (1) where n is the index of the quantized subbands, e is the electron charge, m∗ z is the effective mass along the out-of- plane direction. From the experimental energy splitting of two successive subbands (e.g. E2−E1), Santander et al. could determine the electric field strength F = 83 MV/m and consequently the bottom of the confining potential Vconf(0) = −260 meV, as well as its spatial extension d = 31 ˚A [21]. In the same work, the value of the confining field strength was independently cross-checked by using the Gauss theorem and the carrier den- sity directly measured from the area enclosed by the Fermi surface of the 2DES. Going beyond the simplified but ped- agogical wedge approximation, Meevasana et al. performed Poisson-Schr¨odinger calculations to estimate the band bend- ing profile that acts as a confining potential [22]. The best fit to the experimental data was obtained for Vconf(0) = −700 meV where the lowest subband is localized within ∼20 ˚A from the surface and it is followed by a series of bands with smaller binding energy and deeper extension into the bulk. The cal- culated band bending profile is rather steep as expected for a system with strongly confined two-dimensional states. A band bending profile that extends further into the bulk -as it would be expected for a more three-dimensional system- fails dramatically to simulate the experimental spectra. The two- dimensional character of the observed states can be therefore inferred by the agreement of the experimental ARPES data with the predicted energy dispersion of confined states in a steep electrostatic potential. There are further arguments in favor of the low dimension- ality of the observed states. First of all, the area of the experi- mentally determined Fermi surface consisting of dxy, dxz and dyz orbitals [Figs. 1(e), 1(f)] yields a surface carrier density n2D of the order of 1014 cm−2. If these", + "are further arguments in favor of the low dimension- ality of the observed states. First of all, the area of the experi- mentally determined Fermi surface consisting of dxy, dxz and dyz orbitals [Figs. 1(e), 1(f)] yields a surface carrier density n2D of the order of 1014 cm−2. If these carriers were three- dimensional, the corresponding carrier density n3D would be of the order of 1021 cm−3 a value which is much higher than 3 FIG. 1: (a) The bulk electronic structure of SrTiO3 along ky. dxy, dxz and dyz orbitals give rise to a heavy band along the kz, ky and kx direction, respectively. In the absence of spin-orbit coupling, two light and one heavy band are degenerate at the center of the Brillouin zone (i.e. ky = 0). The bands lie above the Fermi level and they are hence unoccupied. (b) A depth-dependent potential acts as a quantum well that splits each state into a series of subbands. The actual potential is modeled by a triangular quantum well where z = 0 marks the position of the sample surface. (c) Experimental ARPES data showing the effect of the depth-dependent potential on the surface electronic structure of SrTiO3. In contrast to the bulk electronic structure [panel (a)], the bands are shifted below the Fermi level and two dxy subbands are clearly visible. (d) Same as panel (c) using the 2D curvature of the experimental data [30]. The heavy dxz band is also visible. (e) Experimental Fermi surface of SrTiO3(100) consisting of one circular and two elliptical contours that are orthogonal two each other. Dashed lines are guides to the eye and track the Fermi momenta (white and grey circles) acquired by high-statistics measurements at the center of various Brillouin zones. Colors correspond to the orbital origin of the contours following panel (a). (f) Experimental Fermi surface of SrTiO3(100) by plotting the 2nd derivative of the ARPES data. Data have been acquired using 47 eV photons and linear vertical polarization. Part of the figure has been adapted from Ref. 21. the bulk doping of the samples used in Ref. 21. As a mat- ter of fact, the experimental Fermi surface is insensitive to the value of bulk doping, while the latter varies between 1013 cm−3 and 1021 cm−3 [21], thereby pointing towards a surface origin. Moreover, despite their strong dispersion in the kx-ky plane, both studies agree that the observed electronic states do not exhibit any appreciable dispersion along the surface nor- mal, as expected for states with 2D character [21, 22]. From the above, it is evident that the (100) surface of SrTiO3 hosts charge carriers that are not related to the bulk. The presence of a 2DES with high carrier density is therefore well established. We now turn to the origin of the confining potential and the 2DES charge carriers. The two most widely accepted models for the origin of a 2DES on TMO interfaces are the creation of oxygen vacancies [12, 13] and the electronic reconstruction to avoid an interfacial polar discontinuity (i.e. the polar catastro- phe) [11]. The", + "origin of the confining potential and the 2DES charge carriers. The two most widely accepted models for the origin of a 2DES on TMO interfaces are the creation of oxygen vacancies [12, 13] and the electronic reconstruction to avoid an interfacial polar discontinuity (i.e. the polar catastro- phe) [11]. The fact that confined conduction carriers appear at the bare surface of STO(100) which is non-polar cannot be reconciled with the polar catastrophe scenario. In fact, already from the first ARPES studies of a 2DES on STO(100), the role of oxygen vacancies was well established. First of all, there are clear spectral changes in the O2p valence band during the development of the 2DES. An energy shift of its leading edge with respect to the value determined by bulk-sensitive opti- cal adsorption shows that the O2p states are subject to surface band bending [21]. Moreover, gradual loss of the O2p spec- tral weight may be a hint of oxygen vacancy creation at the surface [22]. Loss of spectral weight is observed at the low binding energy (Eb) side of the O2p feature that consists of states which are most likely to change their original environ- ment: Oππ states of non-bonding character or with negligi- ble hybridization with d orbitals [31]. Changes in the valence band are in line with other spectral modifications. Namely, the low-Eb side of the Ti3p peak develops a clear shoulder. This is a fingerprint of Ti ions having lower oxidation num- bers - Ti3+ instead of nominal Ti4+ due to potential oxygen depletion. Moreover, the appearance of the 2DES is accom- panied by another electronic state at an approximate binding energy of 1.3 eV. This state lies in the bulk electronic gap of SrTiO3 and was attributed to oxygen vacancies at the surface 4 FIG. 2: Angle integrated photoemission spectra acquired on a SrTiO3 surface before (red) and after (blue) long exposure to ultra- violet synchrotron radiation. A 2DES (blue spectrum inset) is ac- companied by the generation of an in-gap state related to oxygen vacancies and by spectral changes in the Ti3p peak and the valence band. All SrTiO3 surfaces present the same changes upon UV ex- posure: these data have been acquired on a (111)-oriented surface of SrTiO3. [32]. Figure 2 summarizes the spectral differences between STO surfaces with and without a 2DES. In agreement with expectations from an oxygen-vacancy scenario, Aiura et al. had already shown in 2002 that when the STO(100) surface is exposed to oxygen, the electronic state at the Fermi level is suppressed, the O2p valence band shifts back to lower binding energies and the in-gap state disappears [32]. From the above, it is evident that a high concentration of positively charged oxygen vacancies in the near-surface region is at the origin of the electrostatic potential that bends the conduction band below the Fermi energy (EF) and confines its electrons. As a matter of fact, the electronic structure of the 3d orbitals ap- pears very differently in the bulk and at the surface. In the bulk the t2g conduction band is unoccupied and", + "the electrostatic potential that bends the conduction band below the Fermi energy (EF) and confines its electrons. As a matter of fact, the electronic structure of the 3d orbitals ap- pears very differently in the bulk and at the surface. In the bulk the t2g conduction band is unoccupied and three-dimensional, while on the surface it becomes partially occupied and yields two-dimensional carriers. We finally note that various DFT calculations have confirmed the importance of oxygen vacan- cies in inducing a 2DES at the surface of SrTiO3 [21, 33–35]. Having established oxygen vacancies as the key element for the 2DES at the bare surface of STO(100), another question naturally arises: how are these oxygen vacancies created? An- nealing in UHV during sample preparation can introduce oxy- gen vacancies [36] but this is not the decisive phenomenon for the generation of a 2DES due to many reasons. First of all, we note that a first set of ARPES studies on STO(100) was per- formed on samples that were fractured in UHV rather than surfaces prepared in-situ by annealing [21, 22]. Secondly, oxygen vacancies induced by UHV annealing are distributed rather homogeneously all over the volume of the sample and they do not reside exclusively in the near-surface region [37]. We note that carrier accumulation near the surface is a nec- essary prerequisite in order to have a surface-to-bulk elec- trostatic potential that gives rise to band bending and quan- tum confinement. Finally, an ARPES study by Plumb et al. casted away any remaining doubts on the necessity of high temperature annealing: it was shown that the observed con- fined states are identical for different UHV annealing [38]. Moreover, these states are identical to their counterparts ob- served on cleaved SrTiO3 surfaces [21, 22] and bulk doping has no effect on the near-EF electronic structure [21]. The 2DES seems therefore insensitive to the history of the sam- ple as long as the surface is clean and well-ordered. Oxygen vacancies must be therefore created at later stages. As a mat- ter of fact, Meevasana et al. showed that the 2DES charge density starts from low values and increases with the irradia- tion dose [22]. One can naturally assume that ultraviolet light induces oxygen desorption and hence the creation of oxygen vacancies. The advantage of this method for the generation of oxygen vacancies is that one can control the charge carrier density by tuning the excitation source [22], albeit at the price of a non-homogeneous 2DES when the sample is not irradi- ated uniformly. Up to this point all pieces of the puzzle fall well into place for the 2DES observed on the surface of STO(100): · incoming ultraviolet radiation creates oxygen vacancies at the surface · oxygen vacancies provide a high concentration of near- surface positively charged defects and excess electrons · the positively charged defects create a surface-to-bulk electrostatic field and potential · the electrostatic field/potential induces band bending · the steep band bending depth profile can confine any charge carriers · the formerly unoccupied t2g conduction band follows the profile of the electrostatic", + "positively charged defects and excess electrons · the positively charged defects create a surface-to-bulk electrostatic field and potential · the electrostatic field/potential induces band bending · the steep band bending depth profile can confine any charge carriers · the formerly unoccupied t2g conduction band follows the profile of the electrostatic potential and bends be- low EF in the near-surface region, thus giving rise to conduction electrons · these extra conduction electrons are due to the excess electrons from the oxygen vacancies · the conduction electrons see the depth profile of the electrostatic potential as a quantum well along the sur- face normal and they become confined along this direc- tion · as the surface normal is the only direction of confine- ment, conductions electrons are free to move only along a two-dimensional plane: in other words they form a 2DES · the spectroscopic fingerprint of the 2DES are subbands with appreciable energy-momentum dispersion along any direction parallel to the surface plane, but at the same time with well-determined energy values due to quantum confinement along the surface normal 5 The studies that signalled the discovery of a 2DES on STO(100) [21, 22] were however only the spark. Later research built on this knowledge and brought the field into a more profound understanding. We will summarize these results giving emphasis to ARPES studies that followed various research paths: (i) the experimental observation of 2DES characteristics that remained elusive (e.g. many-body interactions, spin polarization, further details on the creation mechanism, on energy dispersion and on dimensionality). (ii) the investigation of STO surfaces with different orienta- tions. (iii) the generation of 2DES on different functional oxides. (iv) the demonstration of a device-friendly fabrication pro- tocol that may bridge the gap between ARPES studies and technological applications. B. 2DES at the bare surface of SrTiO3: towards a more detailed understanding In order to shed more light into the 2DES, the role of irra- diation in creating a metallic surface was further investigated. Plumb and coworkers observed that on increasing irradiation, the size of the experimental Fermi surface (and hence the car- rier density) saturates very quickly, while intensity continues increasing. They proposed that increasing intensity signals a larger portion of the sample becoming metallic and conclude that there are two stable configurations of STO(100): one in- sulating (no irradiation, no 2D carriers, no Fermi surface) and one with a fixed density of metallic 2D carriers [38]. A later work by McKeown Walker et al. showed that there are not only two stable electronic configurations of STO(100) but that n2D can be continuously tuned as a function of the irradiation time (fluence) and the photon energy [39]. The authors sug- gested a mechanism of vacancy creation based on early results on core hole decay via an Auger process [40, 41]. The mecha- nism involves five consecutive steps: (i) an impinging photon creates a Ti3p core hole, (ii) the core hole is filled by O2p elec- trons via an Auger process, (iii) after the removal of electrons O2−can change to O+, (iv) Coulomb repulsion from neigh- boring Ti4+", + "process [40, 41]. The mecha- nism involves five consecutive steps: (i) an impinging photon creates a Ti3p core hole, (ii) the core hole is filled by O2p elec- trons via an Auger process, (iii) after the removal of electrons O2−can change to O+, (iv) Coulomb repulsion from neigh- boring Ti4+ favors O+ desorption, (v) an oxygen vacancy is created. As the first step of the process requires a threshold energy of 38 eV, there is no surprise that high carrier densi- ties can be achieved only for hν > 38 eV. Carrier densities can be subsequently decreased at will via exposure to a par- tial pressure of O2 [39]. As described in a previous section, the formation of the metallic state on irradiation is accompa- nied by changes in O-related spectral features. The intensity changes of O1s and O2p states are however disproportionate [38]. This signifies that a simple chemical doping scenario is not sufficient to describe the transition to a metallic state, as it must be accompanied by orbital or spatial changes in the near-surface region. In the same study, the Plumb et al. revisited the dimension- ality of the 2DES states [38]. They proposed that although the dxy states are essentially 2D, the dxz / dyz states are more 3D- like, albeit with an out-of plane dispersion that differs signif- icantly from bulk expectations. As a result, it was concluded that dxz / dyz electrons penetrate multiple unit cells towards the bulk, while dxy electrons are more confined in the near- surface region. Tight binding supercell calculations by King et al. are in good agreement with this conclusion [42]. It was shown that out-of-plane potential variations act only as a weak perturbation on electronic states that have large hopping amplitudes along the same direction. In other words, out-of- plane orbitals (i.e. dxz / dyz) are less affected by such a po- tential and they therefore retain some 3D character confirming the aforementioned ARPES results. The combined ARPES + tight binding study by King et al. brought into light further details of the electronic band struc- ture of the 2DES. The energy dispersion was shown to be a result of orbital ordering, spin splitting and many body inter- actions [42]. Orbital ordering refers to the energy splitting of the dxy and dxz / dyz states which are degenerate in the bulk. As first shown in the work by Santander et al., lift of degen- eracy and orbital ordering are natural consequences of con- finement along the out-of-plane direction [21]. According to Eq. (1), states with different masses in the out-of-plane direc- tion will acquire different energies on confinement. From the experimental Fermi surface areas one can derive the orbital or- dering polarization P = n(dxy)−n(dxy/yz) n(dxy)+n(dxy/yz), where n stands for the carrier density. In the case of STO(100), P exceeds 30%. Apart from orbital ordering, breaking of inversion symme- try can lift the spin degeneracy of the surface-confined states through a Rashba-Bychkov (RB) spin-orbit interaction [43– 46]. The tight-binding supercell calculations by King et al. suggest that the RB spin splitting", + "carrier density. In the case of STO(100), P exceeds 30%. Apart from orbital ordering, breaking of inversion symme- try can lift the spin degeneracy of the surface-confined states through a Rashba-Bychkov (RB) spin-orbit interaction [43– 46]. The tight-binding supercell calculations by King et al. suggest that the RB spin splitting is too weak to be observed experimentally. It reaches a maximum value of a few meV at the hybridization points of heavy (dominant dxz/dyz charac- ter) and light (dominant dxy character) bands [42]. Both spin angular momentum and orbital angular momentum abruptly change sign across such hybridization gaps. The possibility of a larger RB spin splitting was investigated in later studies and the main results will be reviewed in a following pragraph. On top of orbital ordering and RB spin splitting, many-body effects modify the energy dispersion of the 2DES states [42]. Electron-phonon and electron-electron interactions renormal- ize the dispersion, resulting in a substantial increase of the effective mass at EF. The electron-phonon interaction in the 2DES states of STO has attracted significant scientific interest already be- fore the experimental demonstration of a 2DES on its surface. Meevasana et al. reported moderate electron-phonon coupling and stressed the absence of polarons on the (100) surface of lightly-doped STO [48]. On the contrary, Chang et al. ob- served a peak-dip-hump lineshape in the near-EF electronic states, where the energy range of the hump and its tempera- ture dependence are in good agreement with the formation of a polaron due to electron-phonon interaction [49]. Later stud- ies solved this apparent controversy by demonstrating that the nature of electron-phonon interaction in STO(100) depends heavily on the carrier density of the 2DES (n2D) [50, 51]. Wang et al. showed that there are two main regimes. If n2D is low (i.e. low 1013 cm−2), evenly spaced band replicas are ob- served below the main bands of the 2DES. The replicas are at- tributed to Fr¨ohlich polarons, quasiparticles formed by an ex- cess electron dressed by a polarization cloud. These quasipar- 6 FIG. 3: (left) Near-EF electronic band structure of the 2DES on SrTiO3(100) at low carrier densities (n2D below 1014 cm−2). The data can be compared with the high-n2D case shown in Figs. 1(c) and 10(a). There is a replica band at an approximate binding energy of 130 meV: the fingerprint of Fr¨ohlich polarons. (right) The angle- integrated photoemission spectrum shows the characteristic peak- dip-hump lineshape. Data have been acquired using 47 eV photons and linear vertical polarization. ticles extend over several lattice sites and propagate through the lattice as a free electron with an enhanced effective mass [52]. The existence of coherent polarons implies long-range coupling to a single longitudinal optical phonon branch. Fig- ure 3 is an example of the replica bands on SrTiO3(100) and the peak-dip-hump lineshape. On the other hand, if n2D is high (i.e. high 1013 cm−2 and low 1014 cm−2), electron- phonon coupling becomes weaker and of a different nature. Replica bands are no more observed and there is instead a kink in the energy dispersion of the 2DES states. These are", + "peak-dip-hump lineshape. On the other hand, if n2D is high (i.e. high 1013 cm−2 and low 1014 cm−2), electron- phonon coupling becomes weaker and of a different nature. Replica bands are no more observed and there is instead a kink in the energy dispersion of the 2DES states. These are spectroscopic fingerprints of the suppression of the long-range Fr¨ohlich interaction and a crossover to a short-range electron- phonon interaction [50]. A related study by Chen et al. re- ported the existence of Fr¨ohlich polarons for carrier densities smaller than 7.6 × 1013 cm−2 [51]. At variance with the re- sults by Plumb et al. [38], the authors suggest an increase of n2D with increasing annealing temperature. In such a scenario the existence -or not- of Fr¨ohlich polarons could be decided already during the preparation of a clean surface. In agree- ment with the study by Wang et al. [50], Chen and cowork- ers do not observe Fr¨ohlich polarons at high n2D [51]. They attribute this fact not to a change in the nature of electron- phonon coupling, but to a 2D-to-3D transition. The key for this transition is, according to the authors, changes in the spa- tial distribution of oxygen vacancies at high annealing tem- peratures. Both studies point out the fact that the LAO/STO interface becomes superconducting for low n2D, or in other words in the regime where the electron-phonon coupling at the surface of STO is strongest [50, 51]: an observation that gives a hint of a possible link between electron-phonon inter- action and superconductivity. We now turn to the possibility of large RB spin splitting on the (100) surface of STO. Theoretical studies on the magni- tude of spin splitting agree that the maximum value is obtained at the hybridization point of light and heavy bands [42, 53– 55]. As mentioned previously, the calculated value is of the order of few meV. This is because Ti is a light element and hence it has a very small atomic spin-orbit coupling. Never- theless, the observation of magnetism at the LAO/STO inter- face [15–17, 56] plus magneto-transport data from which spin splittings as large as 10 meV are inferred [57], maintain the hope of a larger RB effect that goes beyond considerations of the atomic spin-orbit coupling. Indeed a collaborative spin- resolved ARPES (SARPES) study at the Swiss Light Source between CSNSM and the Paul Scherrer Insitute (PSI) con- cluded that the dxy bands of the 2DES are spin polarized and exhibit opposite chiralities, as if they were the two counter- part states in a RB model [58]. The peculiar characteristics of these results are the magnitude of the experimental spin split- ting and the unconventional topography of the spin-polarized subbands. The experimental data were modeled by a com- bined RB + Zeeman splitting where the Rashba parameter αR has a value of 1 order of magnitude larger than what is ex- pected for the calculated electric field at the surface of STO (i.e. 83 MeV/m, see subsection A). As mentioned in Ref. 58, this result is surprising and -inevitably- at", + "+ Zeeman splitting where the Rashba parameter αR has a value of 1 order of magnitude larger than what is ex- pected for the calculated electric field at the surface of STO (i.e. 83 MeV/m, see subsection A). As mentioned in Ref. 58, this result is surprising and -inevitably- at odds with the con- clusions of previous studies concerning the origin of the near- EF electronic states: if the dxy subbands were due to a very large spin splitting and not because of electron confinement, the concept of the 2DES may have to be revisited. However, new experimental results [59] did not reproduce these find- ings. McKeown Walker et al. performed SARPES experi- ments using the PHOENEX endstation at the BESSY II syn- chrotron and showed that spin polarization of the 2DES states falls within the noise level of the instrument as it is at least one order of magnitude smaller than the value proposed in Ref. 58. Such large discrepancies call for further experimental efforts. We note that different protocols for the 2DES gener- ation were followed in each experiment: in-situ prepared sur- faces of undoped STO(100) at the SLS, fractured surfaces of lightly-doped STO(100) at BESSY. Measurement conditions in terms of photon energy and polarization were also differ- ent. Despite such differences in clean surface preparation and in other experimental details, the electronic band structure of the 2DES should be essentially identical following the uni- versality suggested in earlier studies [21, 38]. It is therefore surprising to have such large differences in the spin texture and more SARPES experiments are required in this direction. In summary, the STO(100) surface offers a tunable 2DES with high carrier density, mixed dimensionality, orbital ordering, many body interactions and a potentially large spin splitting. It is therefore no surprise that experimental efforts were extended to different surfaces of STO and to other TMOs. C. 2DESs on SrTiO3 surfaces with different orientations In this section we will discuss experimental ARPES results on bare surfaces of STO having different orientations. The motivations for these studies were on one hand the successful experiments on STO(100) and on the other hand the theoret- ical prediction of topological states at (111) and (110) inter- faces of TMOs [60–62]. The concomitant discovery of 2DESs with novel characteristics at the (111) and (110) interfaces of 7 FIG. 4: (a) The Fermi surface of SrTiO3(110) in the surface plane. The contours are two orthogonal ellipses that become enhanced in adjacent Brillouin zones due to photoemission selection rules. (b) The Fermi surface of SrTiO3(111) in the surface plane. (c) The near-EF electronic band structure of SrTiO3(111) around the center of the Brillouin zone. (d) Fermi surface contours of SrTiO3(111) obtained along a plane normal to the surface by varying the photon energy between 67 eV and 120 eV. States are dispersing very weakly confirming their quasi-2D character. In all cases, red lines denote the borders of the surface Brillouin zone. Data shown in panels (a) and (b)/(c) have been respectively acquired using 91 eV and 110 eV photons. Part of the figure has", + "67 eV and 120 eV. States are dispersing very weakly confirming their quasi-2D character. In all cases, red lines denote the borders of the surface Brillouin zone. Data shown in panels (a) and (b)/(c) have been respectively acquired using 91 eV and 110 eV photons. Part of the figure has been adapted from Ref. 47. LAO/STO [63, 64] further fuelled these efforts. In contrast to the STO(100) surface, the (110) surface of STO is highly polar. It therefore spontaneously reconstructs to avoid a diverging potential [11]. The Fermi surface has twofold symmetry and it consists of two orthogonal ellipsoids that correspond to dxy and dxz/dyz orbitals as revealed by polarization dependent ARPES measurements [47, 67] [Fig. 4(a)]. The observed states do not disperse along the out-of- plane direction, being therefore two-dimensional and com- prising a 2DES at the (110) surface of STO [47, 67]. There is no sign of additional periodicity in the electronic struc- ture, which means that the 2DES is not affected by surface reconstructions. We have therefore proposed that it resides in subsurface layers [47]. This conclusion is further supported by first principles calculations that described the diffusion of oxygen vacancies after exposure of the STO(110) surface to synchrotron radiation [67]. Namely, it was found that oxygen vacancies in STO(110) spontaneously migrate to subsurface layers, while they stay on the surface in the case of STO(100). Confinement along the [110] direction modifies heavily the characteristics of band dispersion. In fact, the effective masses are very different along the direction of confinement with re- spect to their bulk values [47] and even depend on the sub- band index n [67]. This is different to the [100] confinement where effective masses are unchanged with respect to the bulk [21, 47]. The Fermi surface is highly anisotropic and the con- stant energy contours are composed of only dxz / dyz orbitals at higher binding energies [67]. This anisotropy is most prob- ably related to the anisotropic transport behavior observed at the (110)-oriented LAO/STO interface [64]. The (111) surface of STO shares a lot of common features with STO(110). First of all, it is also a highly polar surface with the same nominal charge (±4e) as its (110) counter- part. The existence of a 2DES is supported by the absence of out-of-plane dispersion of the near-EF electronic states [Fig. 4(d)], while its insensitivity to surface roughness and recon- structions reveals that the 2DES is localized in subsurface layers [47, 68]. Tight-binding supercell calculations verify that the corresponding 2DES wavefunctions have negligible spectral weight in the topmost layer [68]. The Fermi surface 8 FIG. 5: A cartoon based on tight-binding calculations that shows the evolution of the Fermi surface of SrTiO3(111) on increasing confine- ment. By a rigid energy shift, the outer contour changes gradually into a graphene-like Fermi surface consisting of non-negligible in- tensity only at the corners of the surface Brillouin zone (K points). The middle panel is reminiscent of the Fermi surface of KTaO3(111) (see Fig. 7). presents a threefold symmetry [Fig. 4(b)] consisting of three -dxy, dxz and dyz- ellipsoids.", + "gradually into a graphene-like Fermi surface consisting of non-negligible in- tensity only at the corners of the surface Brillouin zone (K points). The middle panel is reminiscent of the Fermi surface of KTaO3(111) (see Fig. 7). presents a threefold symmetry [Fig. 4(b)] consisting of three -dxy, dxz and dyz- ellipsoids. In contrast to STO(100), there is no degeneracy lift of the dxy and dxz/dyz orbitals on con- finement along the <111> direction [47, 68] [Fig. 4(c)]. This is because of the cubic symmetry of the underlying perovskite lattice: all the t2g orbitals look the same (hence produce bands with identical effective mass) along the <111> direction. An- other common characteristic with the STO(110) surface is that the effective masses of the confined states are modified with respect to their bulk values. As a matter of fact, the (111) sur- face projection of the bulk bands is not equivalent to a (111)- oriented cut of the bulk FS, in contrast to the case of STO(100) [47, 68]. The carrier density of the 2DES formed on the (110) and (111) surfaces of STO is of the order of 1014 cm−2 as cal- culated by the size of the experimental Fermi surface con- tours [47, 67, 68]. While the 2DES observed on STO(110) might be the key to understand the anisotropic transport be- havior of corresponding interfaces [64], the 2DES observed on STO(111) might be the key for introducing topological characteristics in transition metal oxides. The typical per- ovskite structure of ABO3 is simply a buckled 2D honeycomb lattice if a single (111)-oriented bilayer of B cations is con- sidered. Therefore, if the asymmetry in the confinement po- tential is neglected (case of a surface 2DES) or engineered to vanish (artificial heterostructure), a bilayer-confined (111)- oriented 2DES is equivalent to electrons confined in a honey- comb lattice. According to the cornerstone studies of Haldane and Kane & Mele, in the presence of spin-orbit interaction, the honeycomb lattice can give rise to a finite energy gap at the Fermi level and to in-gap topological edge states [69, 70]. Such a gap has never been observed in graphene due to negli- gible spin-orbit interaction but it may become appreciable in bilayers of (111)-oriented TMOs as proposed by several the- oretical studies [60–62, 71]. At present the spatial extension of the 2DES on the STO(111) surface is larger than a single bilayer. In Ref. 47, the minimum spatial extension has been experimentally estimated at 9 layers at the Fermi level. The corresponding spatial extension of the 2DES on the STO(110) is estimated at 6 layers. Fig. 5 is a cartoon showing how in- creasing confinement of the 2DES on the STO(111) surface can give rise to a graphene-like Fermi surface around the K points of the surface Brillouin zone. Epitaxial growth of a (111)-oriented bilayer and deposition of species that can act as electron donors are possible routes to achieve bilayer con- finement of the 2DES. A comparison of the electronic structure of the three STO surfaces gives a further argument in favor of a oxygen vacancy scenario", + "zone. Epitaxial growth of a (111)-oriented bilayer and deposition of species that can act as electron donors are possible routes to achieve bilayer con- finement of the 2DES. A comparison of the electronic structure of the three STO surfaces gives a further argument in favor of a oxygen vacancy scenario at the origin of the 2DES. Differences in the nomi- nal polar charge are not reflected in the surface carrier den- sity n2D. Namely, 2DESs on the (110) and (111) surfaces of STO exhibit the same carrier density as the non-polar (100) surface. If a polar catastrophe scenario was at play [11], one would expect a much higher carrier density for the (110) and (111) surfaces due to their high nominal polar charge (4e). It is therefore clear that 2DES carriers on STO surfaces do not arise as a means of compensating the polar catastrophe but they must have a different origin. Alternate exposure of the STO(111) surface on synchrotron radiation revealed the origin of the 2DES on this surface. At first, ultraviolet radia- tion generates the near-EF 2DES states and all accompanying spectral features comprising the O-vacancy in-gap state, the Ti3+ shoulder and the loss of spectral weight from the O2p valence band [47, 68]. When the surface is subsequently ex- posed to atomic oxygen, 2DES states and all accompanying spectral features disappear [68, 72]. These experiments are a strong proof that the origin of the 2DES on the STO(111) surface is radiation-induced oxygen vacancies. The most important added value from studies on different STO surfaces is that they reveal the possibility of orientational tuning of the confined electronic band structure. In other words, the direction of confinement is the most important factor determining the shape of the Fermi surface, the effec- tive masses of the 2DES states, the orbital ordering and the presence or absence of degeneracies. For instance, both Figs. 4(a) and 4(d) show the Fermi surface along a (110) plane (or equivalent). Nevertheless, the electronic band structure is very different because the direction of confinement changes. These results therefore offer the possibility to tailor the microscopic properties of confined electron systems at the surface of STO and other TMOs by changing the direction of confinement. 9 FIG. 6: (a), (b) Energy-momentum ARPES intensity maps of the 2DES subbands in KTaO3(100) acquired around the center of different surface Brillouin zones. (c) Same as (b) showing the 2nd derivative of the ARPES intensity. Dashed lines are guides to the eye and color denotes bands of different origin. (d) The Fermi surface KTaO3(100) in the surface plane. Dashed contours are guides to the eye where the color corresponds to the bands traced in panel (c). (e) The Fermi surface of KTaO3(100) obtained along a plane normal to the surface by varying the photon energy. States disperse very weakly confirming their two-dimensional character. (f) Angle-integrated photoemission spectrum showing that the generation of a 2DES on KTaO3(100) is accompanied by an in-gap state, as in the case of SrTiO3 surfaces (see Fig. 2). Data have been acquired with 32 eV [panels (a), (d)]", + "the photon energy. States disperse very weakly confirming their two-dimensional character. (f) Angle-integrated photoemission spectrum showing that the generation of a 2DES on KTaO3(100) is accompanied by an in-gap state, as in the case of SrTiO3 surfaces (see Fig. 2). Data have been acquired with 32 eV [panels (a), (d)] and 41 eV photons [panels (b), (c)]. The polarization was linear horizontal. Part of the figure has been adapted from Ref. 65. D. 2DESs on various transition metal oxides: case studies of KTaO3 and TiO2 anatase Another approach to tailor the properties of confined elec- trons at the surfaces of TMOs is to change the oxide itself. In the present section we will review 2DESs generated on clean surfaces of KTaO3 and TiO2 anatase after exposing them to synchrotron radiation. KTaO3 (KTO) is a wide gap insulator with strong spin-orbit coupling. In comparison to STO, spin-orbit coupling (SOC) in KTO is more than one order of magnitude larger because Ta is a heavier element than Ti (5d vs. 3d). Strong spin-orbit interaction in 5d TMOs can be at the origin of unconventional ground states [73–76] and are of great interest in the emerging field of spintronics [77, 78]. ARPES experiments on clean surfaces of KTO(100) demonstrate the existence of electronic states crossing the Fermi level [Figs. 6(a)-6(c)] [65, 79]. Those states can be well reproduced by a tight binding model of bulk bands subject to a strong spin-orbit interaction and surface confinement [65]. The bands yield a closed Fermi surface in the surface plane [Fig. 6(d)] but there is no dispersion along the out-of-plane direction [Fig. 6(e)] [65, 79]. Moreover, the same near-EF states are observed for substrates with bulk doping varying by at least 5 orders of magnitude [79]. These findings signal a 2DES at the (100) surface of KTO. Similarly to STO, the ob- servation of an in-gap state establishes the oxygen vacancies as the origin of the 2DES [65, 79] [Fig. 6(f)]. Nevertheless, unlike the 2DES observed at the surfaces of STO, there is no evolution of the near-EF band structure on exposure to syn- chrotron radiation: the 2DES on KTO(100) exists right from the beginning. It is therefore possible that the polar nature of KTO(100) favors structural rearrangement that may lower the formation energy of oxygen vacancies [79]. The effect of SOC is evident in the details of the electronic band struc- ture. First of all, the orbital character of the t2g bands becomes mixed and confinement introduces an extra offset of the heavy 10 FIG. 7: (a) The experimental Fermi surface of KTaO3(111) in the surface plane. Dashed contours are approximative guide to the eyes showing the formation of an electron gutter between two contours of triangular symmetry. The results consist of a superposition from data acquired using 96 eV and 50 eV photons. A schematic version of the Fermi surface contours obtained on KTaO3(111) is also shown in the central panel of Fig. 5. (b) Experimental electronic structure along the direction marked by a dashed blue arrow in panel (a). Red dashed contours are parabolic fits", + "using 96 eV and 50 eV photons. A schematic version of the Fermi surface contours obtained on KTaO3(111) is also shown in the central panel of Fig. 5. (b) Experimental electronic structure along the direction marked by a dashed blue arrow in panel (a). Red dashed contours are parabolic fits to the experimental ARPES data. Data in panel (b) have been acquired with hν =96 eV. Part of the figure has been adapted from Ref. 66. bands with respect to the light ones [65]. An additional band of very heavy mass just below EF [Fig. 6(c)] cannot be repro- duced by theory. Taking into account that the cleaved surface consists of patches of (TaO2)1+ and (KO)1−, this band may be attributed to electrons released by the oxygen vacancies in the KO terminations [65]. Comparing the properties of the 2DESs generated on STO(100) and KTO(100), we note that they both have carrier densities of the order of low 1014 cm−2. Following a similar wedge model as in the case of STO(100), the confining potential on KTO(100) has a higher value at the surface (Vconf(0) = −570 meV) and the calculated electric field (F = 280 MV/m) is almost four times stronger than on STO(100) [65]. Moreover, the lower effective masses of the 2DES on KTO(100) could imply higher mobilities [79]. The main message from the electronic band structure of KTO(100) is that mass renormalization and orbital symmetry reconstruc- tion are possible because the values of the spin-orbit coupling, the Fermi energy and the subband splitting become compara- ble [65]. The resulting changes of the electronic band struc- ture have an impact on the properties of the 2DES. The (111) surface of KTO also exhibits a 2DES [66]. In that case, the Fermi surface has threefold symmetry and consists of a network of weakly dispersing “electron gutters” [Fig. 7(a)]. Electron pockets come close to each other near the center of the Brillouin zone [Fig. 7(b)], while the bottom of these bands is non-dispersive along the ΓM direction. This Fermi surface can be well captured by tight binding calculations taking into account electron hopping between consecutive Ta(111) lay- ers [66]. Its area yields a 2D carrier density n2D ∼1014 cm−2. The conduction carriers cannot be three-dimensional because in such a case the corresponding 3D carrier density n3D ∼1021 cm−3 would mean that the KTO substrate is highly-conducting and mirror-like. This is in contrast with the transparent character of the bulk crystal. From the bind- ing energy of the confined states the upper limit of the electric field is estimated at F >166 MeV/m and the minimum bulk extension of the confined states at EF is 7 Ta layers [66]. We note that the observed Fermi surface resembles the simulated FS at an intermediate stage of confinement between STO(111) and a single bilayer (middle panel of Fig. 5). As discussed in the previous paragraphs, a comparison of SrTiO3 and KTaO3 is instructive for the role of the transi- tion metal atom (i.e. B) in the surface electronic properties of ABO3 perovskite TMOs. We will now turn", + "confinement between STO(111) and a single bilayer (middle panel of Fig. 5). As discussed in the previous paragraphs, a comparison of SrTiO3 and KTaO3 is instructive for the role of the transi- tion metal atom (i.e. B) in the surface electronic properties of ABO3 perovskite TMOs. We will now turn to the 2DES observed on the surface of a binary TMO: TiO2 anatase. Sim- ilarly to perovskite TMOs, anatase is a transparent insulator whose crystal structure consists of oxygen octahedra around the transition metal cation (Ti4+). There are however dif- ferences with respect to perovskites: the unit cell is body centered tetragonal, the octahedra are distorted and they are stacked by sharing their edges instead of their corners [26]. ARPES measurements on the (100) and (101) surfaces of anatase show the presence of two bands that form tubular Fermi surfaces having closed contours in the surface plane FIG. 8: The experimental electronic band structure of the (100) [panel (a)] and the (110) surface [panel (b)] of TiO2 anatase around the center of the surface Brillouin zone and along the <010> di- rection. One can readily see two parabolic contours that correspond to confined subbands of dxy origin. The photon energy was 47 eV. More details in Ref. 26. 11 FIG. 9: (a), (d) Fermi surface maps in the surface plane measured on cleaved surfaces of anatase. Panels (a) and (d) correspond to the (100) and (110) surfaces, respectively. Data is the 2nd derivative of ARPES intensity. (b), (e) The bulk Brillouin zone of TiO2 anatase. Green arrows track the relevant high-symmetry directions, while blue and red contours show the relevant high-symmetry planes. Panels (b) and (e) correspond to the (100) and (110) surfaces, respectively. (c), (f) Fermi surface maps obtained along a plane normal to the surface of anatase by varying the photon energy. Panels (c) and (f) correspond to the (100) and (110) surfaces, respectively. Data is the 2nd derivative of ARPES intensity. The negligible out-of-plane dispersion proves the two-dimensional character of the electronic states. Experimental results in panels (a) and (d) have been acquired with 47 eV photons. More details in Ref. 26. and negligible out-of-plane dispersion (Figs. 8 and 9) [80]. This Fermi surface topography is characteristic of confined states. The observed light bands correspond to dxy orbitals as revealed by polarization dependent ARPES measurements [80], in close similarity to STO. However, in contrast to the latter, there are no other bands in the occupied part of the spectrum (Fig. 8). The absence of occupied heavy bands in anatase is due to its tetragonal crystal structure that forces a lift of the bulk t2g degeneracy. As in STO, the 2DES in anatase is fully developed only after sufficiently long exposure to ultra- violet synchrotron radiation. The development of a 2DES is accompanied by all spectroscopic fingerprints that are typical of the creation of oxygen vacancies: in-gap vacancy-related non-dispersive state, Ti3+ shoulder [26, 80]. Hence, the (100) and (101) surfaces of TiO2 anatase host 2DESs that are gener- ated by photo-induced oxygen vacancies. The density of con- duction carriers in", + "2DES is accompanied by all spectroscopic fingerprints that are typical of the creation of oxygen vacancies: in-gap vacancy-related non-dispersive state, Ti3+ shoulder [26, 80]. Hence, the (100) and (101) surfaces of TiO2 anatase host 2DESs that are gener- ated by photo-induced oxygen vacancies. The density of con- duction carriers in anatase as inferred from the corresponding Fermi surface areas are n2D(100) = 5.4 × 1013 cm−2 and n2D(101) = 1.5 × 1014 [80]. The energy-momentum dispersion of the dxy states shows signs of appreciable electron-phonon interaction [26, 81]. This interaction can be tuned with the density of the con- duction carriers as it has been extensively shown in Ref. 81. As in the case of STO(100) [50, 51], at low densities, one observes long-range coupling and well-defined polarons. On the other hand, when the density of conduction carriers becomes higher, polarons lose coherence and dissociate into an electron system coupled to the phonons [81]. We note that the study by Moser et al. reported only one dxy subband that formed an elongated but closed FS contour in the out-of-plane direction [81]. Therefore, the authors concluded that the dxy states have pure bulk character in strong disagreement with the data presented in Figure 9. Discrepancies could be due to the shallow potential well in Ref. 81 that is unable to confine the conduction electrons. However, other unspecified experimental reasons might also affect the results. In contrast to anatase, there is no 2DES on the (110) surface of rutile even in the presence of oxygen vacancies [26]. As rutile is another polymorph of TiO2 consisting of oxygen octahedra with a different stacking order, we 12 conclude that structural factors play an important role in the promotion of conduction carriers from oxygen vacancies [26]. E. A new versatile technique of generating 2DESs on transition metal oxides From the previous sections, it is clear that confined electron systems in TMOs possess fascinating properties that can be modified at will by changing the surface carrier density, the direction of confinement and the underlying oxide. However, with the perspective of realizing functional devices based on oxide electronics, there are important fundamental challenges that had not been tackled in experimental studies of 2DESs on TMO surfaces or interfaces. Specifically, two-dimensional electron systems on inter- faces require an overlayer (e.g. LaAlO3) of thicknesses larger than 15 ˚A in order to show non-zero conductivity [8]. More- over, the growth procedure of the substrate/overlayer system involves deposition techniques (e.g. pulsed laser deposition) which are complex, expensive and result in properties of the 2DES that are largely dependent on all growth parameters [4, 8, 83]. The onset of conductivity and the growth com- plexity make 2DESs observed on LAO/STO and related in- terfaces unsuitable for mass production of oxide-based elec- FIG. 10: (a) Near-EF electronic structure of SrTiO3(100) after the deposition of 2 ˚A of aluminum acquired using 47 eV photons and linear vertical polarization. The inset shows the kink on the band structure due to short-range electron-phonon interaction. (b) Same as (a) using 90 eV photons and linear horizontal polarization. (c) In-plane", + "Near-EF electronic structure of SrTiO3(100) after the deposition of 2 ˚A of aluminum acquired using 47 eV photons and linear vertical polarization. The inset shows the kink on the band structure due to short-range electron-phonon interaction. (b) Same as (a) using 90 eV photons and linear horizontal polarization. (c) In-plane Fermi surface of SrTiO3(100) after the deposition of 2 ˚A of aluminum acquired using 47 eV photons and linear vertical polar- ization. (d) Same as (c) using 90 eV photons and linear horizontal polarization. Experimental conditions in panels (a)/(c) and (b)/(d) enhance light and heavy bands, respectively. Part of the figure has been adapted from Ref. 82. tronic devices. The discovery of 2DESs on the bare surfaces of TMOs did not answer all these challenges. Although the 2DES is not buried under an overlayer, the onset of conduc- tivity is now due to the fact that UV irradiation is essential to create and control the 2DESs [22, 47, 68]. Moreover, the strong variation of the 2DES states as a function of the photon fluence received (number of photons per time per area), means that the spatial variation of the resulting 2DES depends on the size/profile of the beam spot and it can be therefore non- homogeneous. Finally, the absence of an overlayer, although favorable for the use of surface sensitive spectroscopic tech- niques, is an inherent disadvantage for envisaging functional devices because there is no passivation in ambient conditions. All in all, for two-dimensional electron systems in both TMO interfaces and surfaces there exist fundamental challenges that have hampered their functionalization, including an onset of conductivity (interfaces and surfaces), the complicated nature of fabrication techniques (interfaces), the lack of homogene- ity (surfaces) and the absence of a passivation layer (surfaces). Such challenges were signalling the need for a different ap- proach in fundamental experiments before the idea of func- tionalization might be further pursued. Our team has successfully tackled all the above challenges in a very simple way. We demonstrated that 2DESs can be generated on surfaces of TMOs by room temperature deposi- tion of aluminum [82]. Aluminum, an elementary reducing agent, pumps oxygen from the TMO substrate thereby oxi- dizing into insulating AlOx. Most importantly, the formation of oxygen vacancies that are left behind gives rise to confined charge carriers, which are at the origin of the 2DES. The func- tion of deposited aluminum is therefore twofold: on one hand it generates a homogeneous 2DES without the need of UV ir- radiation, while on the other hand it creates an oxide overlayer which acts as a passivation layer for the 2DES. As a matter of fact, having a homogeneous 2DES that requires no external trigger for its existence and is passivated with an insulating overlayer means that the aforementioned fundamental obsta- cles for functionalization are no longer present. On top of that, these achievements were accomplished with the thermal evap- oration of pure aluminum, an approach that is straightforward, versatile and extremely cost-effective. The ARPES results of Figure 10 show the electronic states of the 2DES generated on the (100) surface", + "cles for functionalization are no longer present. On top of that, these achievements were accomplished with the thermal evap- oration of pure aluminum, an approach that is straightforward, versatile and extremely cost-effective. The ARPES results of Figure 10 show the electronic states of the 2DES generated on the (100) surface of SrTiO3 through the deposition of Al. The confined electronic states present sharp energy-momentum dispersions near EF and high-quality Fermi surface contours. There are no spectro- scopic differences with respect to the 2DESs generated by UV light proving that we have an effective -and superior- alterna- tive to irradiation for the generation of a confined electron sys- tem. High-quality 2DESs through the deposition of Al have been successfully generated on other TMO surfaces such as SrTiO3 (111) and anatase (100) [82]. However, the use of Al/TMO interfaces for the creation of 2DESs goes beyond a high-quality alternative to irradiation. As already mentioned, our study has shown that the 2DES is well passivated by an induced Al2O3 layer as long as the deposition of Al reaches a critical value [82]. This critical thickness of the deposited Al corresponds to the generation of an Al2O3 layer of thickness 13 FIG. 11: (a) In-plane Fermi surface of a (100)-oriented BarTiO3 thin film after the deposition of 2 ˚A of aluminum. Data is acquired using 80 eV photons and linear vertical polarization. (b) In-plane Fermi surface of a (100)-oriented BaTiO3 thin film after the deposition of 2 ˚A of aluminum. Data is acquired using 47 eV photons and linear vertical polarization. (c) Near-EF electronic structure of a (100)-oriented BarTiO3 thin film after the deposition of 2 ˚A of aluminum. Data is acquired using 80 eV photons and linear vertical polarization. (d) Near-EF electronic structure of a (100)-oriented BarTiO3 thin film after the deposition of 2 ˚A of aluminum. Data is acquired using 47 eV photons and linear vertical polarization. Experimental conditions in panels (a)/(c) and (b)/(d) enhance heavy and light bands, respectively. Part of the figure has been adapted from Ref. 82. equal its natural thickness on the surface of aluminum (∼1.2 nm). In this case, the 2DES electronic states are still visible in ARPES even after exposure to ambient conditions [82]. Another advantage of this new preparation method is the possibility to create metallic 2DESs on TMO surfaces that do not exhibit confined electron systems upon irradiation. A prime example is the (100) surface of BaTiO3 (BTO). BTO is a well-known ferroelectric material. This ferroelectricity may coexist with metallic behavior up to a critical carrier density, when the metallic region is spatially separated from the ferro- electric one (e.g., a 2DES at the surface vs. a bulk insulating ferroelectric). Using the redox reaction of Al we succeeded in generating oxygen vacancies and hence a confined electron system on BTO(100) [82]. The 2DES/BTO system is equiv- alent to a metal/ferroelectric interface. Hence, polarization switching of the ferroelectric bulk might be used to control the surface conduction as in devices based on ferroelectric re- sistive switching [55, 84]. Fig. 11 shows the spectroscopic fingerprints of near-EF electronic", + "electron system on BTO(100) [82]. The 2DES/BTO system is equiv- alent to a metal/ferroelectric interface. Hence, polarization switching of the ferroelectric bulk might be used to control the surface conduction as in devices based on ferroelectric re- sistive switching [55, 84]. Fig. 11 shows the spectroscopic fingerprints of near-EF electronic states on the (100) surface of BTO. The Fermi surface consists of two orthogonal ellipses and a circular contour. Unlike STO(100), the elliptical con- tours extend beyond the borders of the surface Brillouin zone. The bands are rather broad and it is not clear whether they ac- tually cross the Fermi level or it is the tail of non-negligible spectroscopic weight that gives rise to an apparent Fermi sur- face. In any case, these states have no counterpart in bulk band structure calculations and they appear only after the de- position of aluminum on the (100) surface of BTO. Details of their dispersion and dimensionality might be the subject of future studies. The proposed new method is versatile and gives new perspectives regarding the possible combinations of reducing agents and TMO surfaces. Looking back to the Al/SrTiO3 interface, neither the AlOx capping oxide nor the SrTiO3 substrate possess properties (e.g. ferromagnetism, supercon- ductivity, non-trivial topologies) that can potentially enhance the functionalities of the created 2DES/TMO/capping-oxide system. The exploration of “active” substrates (e.g. ferro- electric BaTiO3) and reducing agents opens new directions of research in the field of confined electron systems on oxide surfaces. IV. Conclusions The field of confined electron systems on the surface of transition metal oxides is no longer in its infancy. The ground- breaking discovery of conducting electron systems on the bare surface of SrTiO3(100) was soon followed by the establish- ment of photo-induced oxygen vacancies as the key element behind the generation of the conduction carriers. Later stud- ies revealed that confined electron systems on transition metal oxides offer an exciting playground where orbital ordering, electron-phonon interactions and possible giant spin splitting coexist. ARPES measurements have played a central role in the development of the story, since they can give direct access to the electronic structure that arises from the confined elec- trons. One can therefore say that not only the 2DESs on TMO 14 surfaces have evolved into an independent scientific field, but also the acquired ARPES knowledge can give insights on the electronic structure of the 2DESs found on TMO interfaces: a neighboring field where spectroscopic techniques with high surface sensitivity have limited hope for success. Such confined electron systems are not limited to the (100) surface of SrTiO3. Different surface orientations and sub- strates have permitted a detailed engineering of the confined electronic structure with respect to a custom-tailored orien- tational tuning [(110) and (111) surfaces], strong spin-orbit interaction (KTaO3 surfaces) and different crystal symmetries (TiO2 anatase). This -so far- ARPES-driven scientific field has been recently become very promising for further studies in ambient conditions with a view to device fabrication. Confined elec- tron systems can be simply generated by the deposition of an elemental reducing agent on the surface of a transition metal oxide.", + "(TiO2 anatase). This -so far- ARPES-driven scientific field has been recently become very promising for further studies in ambient conditions with a view to device fabrication. Confined elec- tron systems can be simply generated by the deposition of an elemental reducing agent on the surface of a transition metal oxide. The major advantage of this fabrication technique is that the generated 2DES is capped by the natural oxide of the reducing agent. Functional devices based on confined electron systems on multifunctional oxides (ferroelectric or ferromagnetic) may be therefore studied in ambient pressure due to an oxide capping layer. This simple, cost-effective and versatile technique gives rise to spatially homogeneous elec- tron systems and overcomes previous limitations of 2DESs generated at TMO interfaces and by UV irradiation. 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Hutchinson,1 M. Baker1 and F. Marsiglio1,2,∗ 1 Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2E1 2Physics Division, School of Science and Technology University of Camerino, I-62032 Camerino (MC), Italy (Dated: June 4, 2018) The two electron configuration in the Helium atom is known to very high precision. Yet, we tend to refer to this configuration as a 1s ↑1s ↓singlet, where the designations refer to Hydrogen orbitals. The high precision calculations utilize basis sets that are suited for high accuracy and ease of calculation, but do not really aid in our understanding of the electron configuration in terms of product states of Hydrogen orbitals. Since undergraduate students are generally taught to think of Helium, and indeed, the rest of the periodic table, in terms of Hydrogenic orbitals, we present in this paper a detailed spectral decomposition of the two electron ground state for Helium in terms of these basis states. The 1s ↑1s ↓singlet contributes less than 93% to the ground state configuration, with other contributions coming from both bound and continuum Hydrogenic states. I. INTRODUCTION As early as 1928 Hylleraas1,2 recognized that using Hydrogenic orbitals to describe the ground state elec- tron configuration of the Helium atom was not a good idea. In the introduction to his seminal paper he says, “It thereby appeared that the use of hydrogen eigenfunc- tions, as done by Dr. Biem¨uller, leads to erroneous re- sults, which obviously has to do with the fact that they do not form a complete functional system.”1 He then goes on to introduce a set of coordinates and variational wave functions that more accurately describe the two-electron ground state of the Helium atom.1,3 To our knowledge, the results of Dr. Biem¨uller were never published. Most students nowadays, if asked about the electronic ground state configuration of the Helium atom, will respond that it is the singlet state4 of 1s ↑1s ↓, as listed, for example, in many periodic tables. Even more senior colleagues, while realizing that this simple characterization omits interactions, become disturbed at the suggestion that other orbitals enter the ground state configuration, since this appears, at first glance, to be at odds with the ‘closed-shell’ character of He, and the fact that it is an ‘inert’ element. This misunderstanding tends to be reinforced in the way we teach undergradu- ate quantum mechanics: since the two-electron problem is already very difficult, Helium is often used as an exam- ple case study for many approximate methods, including perturbation theory and the variational method. Usu- ally only corrections to the energy (and not the wave function) are described, and even with the variational method, often a simple extension of a (singlet) 1s ↑1s ↓ is used. This tends to reinforce the incorrect idea that the singlet 1s ↑1s ↓Hydrogenic product wave function is the entire picture.5 The solution to this problem attained by Hylleraas was to adopt a set of basis functions that more accurately capture the electron", + "of a (singlet) 1s ↑1s ↓ is used. This tends to reinforce the incorrect idea that the singlet 1s ↑1s ↓Hydrogenic product wave function is the entire picture.5 The solution to this problem attained by Hylleraas was to adopt a set of basis functions that more accurately capture the electron correlations that exist in the Helium two-electron ground state. We want to emphasize here that there is no question that this works extremely well.6 Nonetheless, physicists and chemists still tend to think of atomic electronic configurations in terms of Hydrogenic orbitals (the periodic table guides us in this direction), and this is how our intuition is formed. Thus, in spite of the lesson of Dr. Biem¨uller’s (erroneous) calculation, we think it is important to answer the question, ‘Just how much of the Helium two-electron ground state consists of the singlet 1s ↑1s ↓configuration?’ and, as an obvious follow-up, ‘What other Hydrogenic states contribute to the ground state configuration?’7 Most (non-expert) col- leagues are shocked when the answer requires continuum states.8 Aside from pedagogical value, the requirement of more than one electronic configuration has also become a po- tentially important ingredient in models of metals and superconductors within the field of condensed matter.9–11 The electronic ground state of Helium has served as the ‘poster child’ for atoms in a solid state environment, where the electron occupation can fluctuate. An impor- tant ingredient in electron conduction is the fact that most lattice models in condensed matter physics focus on one set of orbitals from each atomic site; these or- bitals overlap to form a conduction band, and the band that crosses the Fermi energy, i.e. the energy separating occupied from unoccupied bands, is the one band of in- terest for low energy excitations. But this reasoning is primarily based on single electron ideas, and as a rem- edy, this model (tight-binding, or H¨uckel) is often gener- alized to include electron interactions with one another, especially when two electrons occupy the same orbital on the same site (the so-called Hubbard model12). The Hubbard model, and, in general, Hubbard-like models, are the subject of intensive investigation in condensed matter.13 However, almost all these models miss the im- portant ingredient that the same single electron orbitals that are pertinent for single electron occupation are not adequate to describe a doubly occupied situation. That is, the situation, “when two electrons occupy the same or- bital on the same atom” is simply not possible, without a modification of those orbitals.14 In reality, two electrons that find themselves on the same atom will spread out 2 to avoid one another, and therefore inevitably occupy (at least partially) other single particle orbitals (like the 2s and 3s orbitals in the case of Helium). The present study of Helium, while avoiding more complicated ideas like Wannier bases and Hartree-Fock orbitals, will emphasize this very point. As such, the two-electron configuration in the Helium atom serves as a stepping stone between a standard quantum mechanical ‘textbook’ problem and a more research-oriented set of problems. We first outline the general problem of solving for", + "complicated ideas like Wannier bases and Hartree-Fock orbitals, will emphasize this very point. As such, the two-electron configuration in the Helium atom serves as a stepping stone between a standard quantum mechanical ‘textbook’ problem and a more research-oriented set of problems. We first outline the general problem of solving for the eigenstates of two-electrons bound to a central nu- cleus. This will first be done using ‘natural’ basis states consisting of product states of the bound state hydro- genic states. This is presumably the path followed by Dr. Biem¨uller, but since, to our knowledge, this exer- cise has never been published, we provide an outline of the method, with most of the details relegated to an ap- pendix. As already remarked, this procedure does not succeed in converging to the known answers for the He- lium atom, for the reason that the continuum states are required, and we arrive at the surprising (for some) con- clusion that an accurate description of the Helium ground state cannot be provided by using just the Hydrogenic bound states. We thus turn to a more direct approach, where, start- ing with the exact wave function, one can compute the coefficients of the various basis states of interest. For He- lium, while the exact ground state wave function is known numerically to many significant digits,6 we instead use the so-called Hylleraas wave function,3 consisting of only three variational parameters,15 in the interest of simplic- ity and transparency. This will serve as our ‘exact’ wave function, and, consistent with what we stated above, the sum of the magnitudes of the coefficients of the basis states consisting of product Hydrogenic orbitals will fall short of unity, with the remainder coming from the con- tinuum states. While much of this discussion is suited for undergrad- uates, some of the mathematics is more suited for un- dergraduate projects and/or graduate courses. In either case we feel that this approach to the problem provides a useful connection between course work and research-type problems. We have tried to keep much of the mathemat- ics (which we have done in part analytically and in part with software packages like Mathematica or Maple) in Appendices. II. THE HELIUM TWO ELECTRON PROBLEM A. Preliminaries The energetics of the two electron Helium atom are very well known.6 Here we concern ourselves with just the non-relativistic interactions, so the Hamiltonian gov- erning this system is H = −¯h2 2m (∇2 1 + ∇2 2) −Ze2 4πǫ0 \u0012 1 r1 + 1 r2 \u0013 + e2 4πǫ0 1 r12 (1) where r12 ≡|⃗r1 −⃗r2| is the separation between the two electrons, and r1 ≡|⃗r1| (r2 ≡|⃗r2|). Note that we have al- ready used an important approximation in writing down Eq. (1) — we have adopted the Born-Oppenheimer ap- proximation, which essentially assumes that the mass of the nucleus is infinite. Hence the only degrees of freedom are those of the two electrons, each with charge −e and mass m. The factor of Z in the middle term is present because in general the charge of the nucleus is Ze;", + "which essentially assumes that the mass of the nucleus is infinite. Hence the only degrees of freedom are those of the two electrons, each with charge −e and mass m. The factor of Z in the middle term is present because in general the charge of the nucleus is Ze; obvi- ously for Helium Z = 2. It is the last term, representing the electron-electron repulsion, that causes the difficulty; this is the term which is often ignored (at least concep- tually) in a student’s first exposure to the Helium atom (and the periodic table, for that matter) in his/her un- dergraduate education. The usual procedure is to first ignore the electron- electron repulsion; the problem is then readily solved since it now consists of essentially a Hydrogen problem that has to accommodate two electrons. The ground state solution is then ψ1(⃗r1, ⃗r2) = φ100(⃗r1)φ100(⃗r2), (2) which is a product state of two single electron solutions,16,17 φnℓm(⃗r) = Rnℓ(r)Y m ℓ(θ, φ), (3) where (nℓm) are the usual quantum numbers, and R(r) [Y m ℓ(θ, φ)] is the radial [angular] part of the wave func- tion, which can be written in terms of Associated La- guerre polynomials and Spherical Harmonics, respec- tively. These functions are tabulated in most texts; for reference we write down the ground state result needed in Eq. (2): φ100(⃗r1) = 2 \u0000 Z a0 \u00013/2e−Zr1/a0 1 √ 4π (4) where a0 is the Bohr radius and it is understood that Z = 2 for Helium, but we have left it general in Eq. (4). As it stands, Eq. (2) is deceptively simple; we have left out the spin degree of freedom, and therefore we have omitted any discussion of parahelium states (symmet- ric spatial and antisymmetric spin wave function compo- nents) and orthohelium (antisymmetric spatial and sym- metric spin wave function components). In the absence of spin-orbit coupling these two classes of states are not coupled by the Hamiltonian (1), and one can focus on one or the other. Since we are interested in the ground state configuration, we will focus on the parahelium states — these contain the lowest energy basis state and therefore the ground state. This means that all the two particle states that we consider below should be understood to 3 include the singlet spin state, |χ(⃗s1,⃗s2)⟩= 1 √ 2 \u0002 | ↑⟩1 | ↓⟩2 −| ↓⟩1 | ↑⟩2 \u0003 , (5) where the subscripts 1 and 2 refer to particle 1 and 2.18 From this point on spin is no longer included in the dis- cussion, but its presence has indeed dictated that we treat only symmetric orbital states, of which the state (2) is one. Other examples are listed in the Appendix, and will be used below. B. Matrix Mechanics Because this approach was so successful in other prob- lems, our first line of attack is to utilize matrix mechan- ics; we decompose the ground state into a complete set of simple well known basis states, |ψ⟩= ∞ X i=1 ai|ψi⟩ (6) where the ai’s are the unknown coefficients. Inserting this into", + "approach was so successful in other prob- lems, our first line of attack is to utilize matrix mechan- ics; we decompose the ground state into a complete set of simple well known basis states, |ψ⟩= ∞ X i=1 ai|ψi⟩ (6) where the ai’s are the unknown coefficients. Inserting this into the time-independent Schr¨odinger equation, and taking inner products with the same basis states yields the eigenvalue equation, ∞ X j=1 Hijaj = Eai, (7) where the matrix elements are given by Hij = ⟨ψi|H|ψj⟩. (8) As already remarked, a complete basis set of Hydrogenic orbitals consists of an infinite number of bound states (of which Eqs. (A1a-A1f) are the first few), and an infinite continuum of unbound states, for which a much more complicated expression is required (though it is essen- tially the analytical continuation of the bound states). But our tact will be to forge ahead, and simply truncate the expansion to include only the low-lying bound states, thus excluding a (infinite) number of bound states and all the continuum states. Such a truncation scheme worked extremely well for the harmonic oscillator,19 where, in the square well basis used for that problem, only the first ten or so states were required to give fully converged results. One calculational advantage of this choice of basis states is that the one electron parts of the Hamil- tonian [all but the last term in Eq. (1)] return eigenvalues of the Hydrogen spectrum when operat- ing on these states. That is, focusing on a product, φn1,ℓ1,m1(⃗r1)φn2,ℓ2,m2(⃗r2), which covers the most general case encountered in Eqs. (A1a-A1f), we have Hφn1,ℓ1,m1(⃗r1)φn2,ℓ2,m2(⃗r2) = \u0002 −Z2E0 \u0000 1 n2 1 + 1 n2 2 \u0001 + ˆHint \u0003 φn1,ℓ1,m1(⃗r1)φn2,ℓ2,m2(⃗r2), (9) where ˆHint = e2 4πǫ0 1 |⃗r1 −⃗r2| (10) is the “hard part”, i.e. the electron-electron interaction potential. In Eq. (9) and below we use Z = 2 and E0 ≡¯h2/(2ma2 0) ≈13.606 eV. To consider an actual matrix element between any general two electron states, we require twelve quantum numbers; hence to avoid pro- liferation of indices we use the shorthand i1 ≡n1ℓ1m1. Then ⟨φi1φi2|H|φi3φi4⟩= −Z2E0δi1,i3δi2,i4 \u0000 1 n2 1 + 1 n2 2 \u0001 + ⟨φi1φi2|Hint|φi3φi4⟩, (11) where the last matrix element requires more careful anal- ysis; details of the calculation of this matrix element, including some examples, are provided in Appendix A. There a number of states are listed [see Eqs. (A1a-A1f)], and a software package like Maple or Mathematica can be tasked with evaluating all the required matrix elements up to some cutoff, imax; the resulting finite matrix can be easily diagonalized, and we can obtain the corresponding ground state eigenvalue and eigenvector. C. Results In Fig. 1 we show the ground state energy as a function of the inverse of imax, with square symbols, along with a guide to the eye; it is clear that the energy has essen- tially saturated as imax increases, but to a value much higher than the known exact value, which is indicated by the large (red) square at the origin. The reason for this discrepancy has already been", + "along with a guide to the eye; it is clear that the energy has essen- tially saturated as imax increases, but to a value much higher than the known exact value, which is indicated by the large (red) square at the origin. The reason for this discrepancy has already been noted: including only the bound states is not sufficient — they do not form a complete basis set, and the contribution from the contin- uum states is significant. At first glance it may seem odd that continuum states contribute to the ground state for the Helium atom; however, one must keep in mind that (i) the right combination of plane waves, for example, can indeed describe a very localized state, and (ii) with a poor choice of basis states (which, indeed, the Hydro- genic states are), the bound states will simply not be sufficient to describe the Helium atom. This latter point is important, as a truncation in the (infinite) basis for the problem in Ref. (19) did not deteriorate the solution there, as the set of basis states used in that reference were sufficiently ‘good’ to describe the problem at hand. Later we shall demonstrate the role of continuum states in this problem. Also shown in Fig. 1 is the probability for the singlet (1s ↑1s ↓) basis state (right hand scale) as a function of 1/imax (blue asterisks). The horizontal line indicates the saturated value, about 91%, indicating that a significant fraction (close to 10%) comes from other contributions. However, because of the problem mentioned above, this 4 −80 −79 −78 −77 −76 −75 −74 0.0 0.2 0.4 0.6 0.8 1.0 Energy (eV) 1/imax 0.0 0.5 1.0 (a1)2 FIG. 1. (color online) Results for the ground state energy (filled green squares) as a function of imax (see text and Ap- pendix A for a full explanation of imax). The curve provided is merely a guide to the eye. Notice, however, that the ex- trapolated result to the origin (as imax →∞) is well above the known exact result, indicated with the large (red) square at −79.0 eV. The discrepancy is explained in the text. Also shown are blue asterisks, which indicate the probability of the (1s ↑1s ↓) basis state (right hand scale). It saturates at approximately (a1)2 ≡a2 (100,100) ≈0.91, through which a horizontal (blue) line is drawn, indicating that a significant amplitude comes from other states. value is not reliable, and we will tackle this issue in the next section. To summarize this section, we have attempted to de- scribe the configuration of the two-electron ground state in Helium, using only the bound state Hydrogenic or- bitals. The reason for doing this is that these orbitals are the ones with which we have the most intuition, and they are the ones we tend to use to gain a preliminary under- standing of the periodic table. Our hope was that, while this set of basis states is infinite, a finite set used through some truncation scheme would capture the essence of the ground state configuration; in reality not only", + "are the ones we tend to use to gain a preliminary under- standing of the periodic table. Our hope was that, while this set of basis states is infinite, a finite set used through some truncation scheme would capture the essence of the ground state configuration; in reality not only did this not work well, it is clear that even if one extrapolates to the use of all the Hydrogenic bound states, this incomplete basis will not work. In fact, the continuum states are required for a proper description. The reason for this is that the continuum states are able to describe high resolution spatial corre- lations that the bound states cannot. The bound states become more extended as their quantum numbers in- crease, and so any finer scale spatial correlations will have to be properly described by the continuum states. This conclusion will be reinforced in the next section, where we examine projections of the various basis states on a very accurate but simple representation of the exact two- electron wave function. III. PROJECTIONS A. Hylleraas wave function: bound states One could try to include the continuum states in the preceding calculation. To avoid an infinite matrix would require a judicious selection of these states, presumably based on their energies. In our opinion this procedure is fraught with difficulties and various choices are possible, so we do not pursue this tact. What we really want is a good estimate of the contribution of the ‘naive’ (1s ↑ 1s ↓) basis state to the actual electronic ground state of Helium. The contributions from the continuum states are of importance only insofar as they influence the value a2 1 ≈0.91 obtained in the previous section. Thus we take a different approach, and find that the conclusion of the previous section is reinforced quantitatively. Using Eq. (6), we can formally rewrite this as Ψexact = ∞ X m=0 am|ψm⟩+ Z dp ap|ψcont p ⟩, (12) where now it is clear that a complete set of states is being used; the |ψm⟩are those enumerated in Eqs. (A1a-A1f) and beyond, while the |ψcont p ⟩refer to the (as yet un- specified) continuum basis states. These states, taken together, in fact form a complete, orthonormal set. Mul- tiplying on the left by ⟨ψn| and forming the inner product therefore projects out the contribution from the nth basis state: an = Z ψnΨexact. (13) For Eq. (13) to be useful we need to know Ψexact; as mentioned in the Introduction, this is known numeri- cally to many digits precision,6,15 using a Hylleraas-type basis:3,20 Ψ = X ci,j,ℓe−ks/2sj uℓti (14) where the Hylleraas coordinates are defined s ≡(r1+r2)/a0, t ≡(r2−r1)/a0, u ≡r12 ≡|⃗r1−⃗r2|/a0 (15) and the summation occurs all over non-negative integers for j and ℓ, and only even non-negative integers for i. This variational basis set has been expanded in a variety of ways, as summarized in Ref. (6), but in what follows we take only three terms from Eq. (14), with (i, j, ℓ) = (0, 0, 0), (i, j, ℓ) = (2, 0, 0), and (i,", + "non-negative integers for i. This variational basis set has been expanded in a variety of ways, as summarized in Ref. (6), but in what follows we take only three terms from Eq. (14), with (i, j, ℓ) = (0, 0, 0), (i, j, ℓ) = (2, 0, 0), and (i, j, ℓ) = (0, 0, 1). Then we define the ‘Hylleraas wave function’ to be the three parameter wave function, ΨHy(⃗r1, ⃗r2) = 2 π \u0012kZ a0 \u00133 e−Zks\u0002 1 + 2Zc1ku + c2(2Zkt)2\u0003 , (16) where k, c1, and c2 are the three parameters to be deter- mined by minimizing the energy, and Z will eventually 5 TABLE I. Results for some overlaps, ai. i basis state ai |ai|2 Tot. Prob. 1 100 100 0.9624 0.9263 0.9263 2 100 200 -0.2148 0.0461 0.9725 3 100 300 -0.0752 0.0057 0.9781 4 100 400 -0.0427 0.0018 0.9799 5 100 500 -0.0289 0.0008 0.9807 6 100 600 -0.0213 0.0005 0.9812 7 100 700 -0.0166 0.0003 0.9815 8 21-1 211 0.0260 0.0007 0.9822 9 210 210 -0.0184 0.0003 0.9825 10 200 200 -0.0146 0.0002 0.9827 11 200 300 -0.0090 0.0001 0.9828 12 100 320 0 0 0.9828 be set to equal 2. Eq. (16) will serve as our ‘exact’ wave function. In reality, minimization of the energy with ΨHy yields EHy ≈−78.979 eV, with variational parameters c1 ≈0.0803 and c2 ≈0.0099, and k ≈0.908. Evaluating the necessary integrals and obtaining these results is well documented,3,20 and so these are simply summarized in Appendix B. The attained energy is within 0.05% of the exact result (−79.014 eV); we consider this sufficiently close to justify our adoption for the present purposes of 1 √ N |ΨHy⟩as the ‘exact’ normalized Helium wave func- tion, with N given below. The calculation of the overlap integrals for the bound states can be done for the general case — some details are provided in Appendix B, along with a simple example. Our results are summarized in Table 1. It is clear that the largest contributions arise from states in which one of the electrons is in the φ100(⃗r) state, i.e. the single electron ground state. In particular, more than 92% comes from the 1s ↑1s ↓singlet, but a sizeable contribution comes from states other than this one. Fur- ther (small) contributions arise from states not listed. While the total probability (5th column in Table 1) is close to unity, inclusion of the remaining bound states (not shown) still gives a total probability that falls short of 0.99. The remaining probability arises from continuum states, to which we turn in the next subsection. B. Overlap integrals: continuum states For Helium, including the full spectrum of continuum two particle states into the problem of computing the ground state configuration is very complicated. Noting that the most important contributions from the bound states arise when one of the electrons is in the one elec- tron ground state (see Table 1), we will likewise compute only the contributions from the continuum states when one of the two electrons is in the one electron (bound) ground", + "that the most important contributions from the bound states arise when one of the electrons is in the one elec- tron ground state (see Table 1), we will likewise compute only the contributions from the continuum states when one of the two electrons is in the one electron (bound) ground state. These states, in the singlet configuration, are written ψp(⃗r1, ⃗r2) = 1 √ 2 \u0000φp(r1)φ100(r2) + φ100(r1)φp(r2) \u0001 (17) where only one label (the momentum p ≡|⃗p|) is required because the other electron is in the 1s state so the con- tinuum state also has ℓ= m = 0, and we have written the right-hand-side of Eq. (17) as depending only on the radial coordinates, r1 and r2. The single particle state φp(r) has a radial part Rp(r) given by Rp(r) = Z a0 s 2π pa0 Z 1 −e−2π Z pa0 e−iprM(1 + i Z pa0 , 2, 2ipr), (18) where we have written this for general Z (here we need only Z = 2), and M(a, b, z) ≡ ∞ X m=0 (a)m (b)m zm m! (19) is the so-called Kummer function,21 and (a)m ≡a(a + 1)(a + 2)...(a + m −1) is the so-called Pochhammer sym- bol. Note that (a)0 ≡1. Equation (18) is essentially the analytical continuation of the radial bound state wave functions. The standard22 normalization condition for the continuum states, Z ∞ 0 dr r2Rp′(r)Rp(r) = δ(p′ −p), (20) determines the coefficient in Eq. (18). Just as for the bound states we require ap = Z d3r1 Z d3r2ψp(r1, r2) 1 √ N ΨHy(⃗r1, ⃗r2). (21) Because of symmetry the contributions from the two terms in Eq. (17) are identical; the integrals involved in Eq. (21) can be done, either with Mathematica or by hand. Some detail is provided in Appendix C. With y ≡pa0/Z the result is 6 ap = r π N 32k3 (k + 1)5 ra0 Z r y 1 −e −2π y \u001a 2(k + 1)2I2 + c1 \u0002 16kI1 + 4k(k + 1)2I3 −4k(k + 1)J2 −16kJ1 \u0003 + c2 \u0002 96k2I2 + 8k2(k + 1)2I4 −48k2(k + 1)I3 \u0003\u001b (22) where the expression for In and Jn are provided in Ap- pendix C. Then the total contribution for these contin- uum states, Pcont = 2 π Z ∞ 0 dp|ap|2, (23) is evaluated numerically; the result is a further contribu- tion of Pcont ≈0.0117, which brings the total probabil- ity from the bound states in Table 1 and the continuum states shown in Eq. (17) to ≈0.995. The remaining prob- ability required to reach unity comes from bound state contributions not included in Table 1 and from contin- uum states beyond those not considered in Eq. (17). IV. SUMMARY The primary purpose of this paper was to demonstrate the degree to which the electron configuration in the He- lium atom is not simply the singlet 1s ↑1s ↓state. In fact we have shown, through two methods, diagonaliza- tion in a particular basis set, and through projection on this same basis, that approximately 8% of the wave func- tion is", + "degree to which the electron configuration in the He- lium atom is not simply the singlet 1s ↑1s ↓state. In fact we have shown, through two methods, diagonaliza- tion in a particular basis set, and through projection on this same basis, that approximately 8% of the wave func- tion is not the 1s ↑1s ↓singlet. As already noted, this has a very large effect on the energy. While we mentioned in the introduction that this fact may be of interest in current research in strongly correlated electron systems, including superconductors, the emphasis here has been on pedagogical aspects. Further exploration of the con- sequences of this electron configuration in a solid can be the topic of specialized student research projects. The diagonalization of the problem in a truncated basis was, in fact, not successful at reproducing the ground state energy; in being unsuccessful, this calculation has served to highlight important pedagogical points, and it is partly for this reason that we have included a detailed analysis of the problem here. The two-electron problem in Helium served to highlight that the bound Hydrogenic states do not form a complete basis set, and while one might have thought that the low-lying states would be sufficient to accurately describe the ground state, this work shows that this supposition is incorrect. Setting up the electron configuration in Helium as a matrix diagonalization problem also serves to provide a concrete example of matrix mechanics, which, for un- dergraduates at least, is often introduced only in a for- mal sense, with an abstract-only exposition of Hilbert space and expansion in basis functions, etc. This prob- lem also serves to provide a first exposure to a realis- tic ‘many-body’ problem, and how one would construct many particle wave functions (here, two electrons is the first stepping stone in this direction beyond the single particle problem to which undergraduates are normally exposed). The second method we presented, projection of an ac- curate wave function onto an orthonormal basis set of ‘known’ and well-understood wave functions, requires a little more sophistication, because Hylleraas wave func- tions are usually only introduced in graduate studies. Nonetheless, the three-parameter Hylleraas wave func- tion given in Eq. (16) is sufficiently simple to be suitable for undergraduates. The ensuing algebra for the bound states is also readily accessible, particularly if projec- tions are computed on a case-by-case basis. For example, evaluation of Eq. (B11) requires knowledge only of ele- mentary functions and standard integrations. We have included projections involving the continuum states as well, and these require more advanced mathematics, and certainly can be skipped at the undergraduate level. The topic of even two electron correlations has been traditionally confined to more advanced studies at the graduate level. Part of the reason for this is that there are better and more accurate methods available after a pre- liminary introduction to many-body methods has been assimilated by the student. However, most students see the electronic structure of atoms in the periodic table in general, and the Helium atom in particular, in terms of Hydrogen orbitals, and therefore", + "there are better and more accurate methods available after a pre- liminary introduction to many-body methods has been assimilated by the student. However, most students see the electronic structure of atoms in the periodic table in general, and the Helium atom in particular, in terms of Hydrogen orbitals, and therefore it is desirable that a de- scription be provided in terms of these same orbitals, as we have presented in this paper. ACKNOWLEDGMENTS We thank Don Page and Bob Teshima for helpful discussions concerning the treatment of the continuum states .This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), by the Canadian Institute for Advanced Re- search (CIfAR), and by a University of Alberta McCalla Professorship and Teaching and Learning Enhancement Fund (TLEF) grant. ∗permanent address for FM is University of Alberta. 7 Appendix A: Matrix mechanics for Helium 1. General considerations The first 6 two particle states that need to be consid- ered [dropping the spin part given in Eq. (5)] are, for example: ψ1(⃗r1, ⃗r2) = φ100(⃗r1)φ100(⃗r2), m1 + m2 = 0 (A1a) ψ2(⃗r1, ⃗r2) = 1 √ 2 \u0002 φ100(⃗r1)φ200(⃗r2) + φ100(⃗r2)φ200(⃗r1) \u0003 , m1 + m2 = 0 (A1b) ψ3(⃗r1, ⃗r2) = 1 √ 2 \u0002 φ100(⃗r1)φ21−1(⃗r2) + φ100(⃗r2)φ21−1(⃗r1) \u0003 , m1 + m2 = −1 (A1c) ψ4(⃗r1, ⃗r2) = 1 √ 2 \u0002 φ100(⃗r1)φ210(⃗r2) + φ100(⃗r2)φ210(⃗r1) \u0003 , m1 + m2 = 0 (A1d) ψ5(⃗r1, ⃗r2) = 1 √ 2 \u0002 φ100(⃗r1)φ211(⃗r2) + φ100(⃗r2)φ211(⃗r1) \u0003 , m1 + m2 = 1 (A1e) ψ6(⃗r1, ⃗r2) = φ200(⃗r1)φ200(⃗r2), m1 + m2 = 0 (A1f) Note that all these states are symmetric, and that we have indicated the quantum number m1 + m2 for each two-particle state. In any state in which this is not zero, that state will not contribute to the ground state. This selection rule will be derived below, along with some other rules that eliminate more of these states. These rules provide a big simplification, and to the degree that we anticipate that maybe the (nℓm) = (600) doesn’t contribute very much to the ground state, the number of states that may be required might be anticipated to be small. Unfortunately, this is not the case, as the continuum states contribute as well; this fact ultimately thwarts this attempt to attain a spectral decomposition in terms of these basis states.3 Nonetheless, we proceed along this line of investigation, first to demonstrate that this is the case, and secondly to establish these helpful selection rules. To evaluate the last matrix element in Eq. (11) we introduce the expansion23,24 1 |⃗r1 −⃗r2| = ∞ X k=0 k X mk=−k (k −|mk|)! (k + |mk|)! rk < rk+1 > P |mk| k (cosθ1)P |mk| k (cosθ2)eimk(φ1−φ2) (A2) where r< = r1 and r> = r2 if r1 < r2 and vice versa for r1 > r2. The P mk k are the associated Legendre poly- nomials, and the angles correspond to the spherical co- ordinates for each of the vectors, ⃗r1 and ⃗r2. Eq. (A2) should become familiar to students through", + "and r> = r2 if r1 < r2 and vice versa for r1 > r2. The P mk k are the associated Legendre poly- nomials, and the angles correspond to the spherical co- ordinates for each of the vectors, ⃗r1 and ⃗r2. Eq. (A2) should become familiar to students through problems in Electromagnetism as well as in Quantum Mechanics. We can expand each of the individual Hydrogenic states (see Eq. (3)) and use Y m ℓ(θ, φ) = ǫm s (2ℓ+ 1) 4π (ℓ−|m|)! (ℓ+ |m|)!eimφP m ℓ(cos θ), (A3) where ǫm = (−1)m for m ≥0, and ǫm = 1 for m ≤0. If we substitute into the last term of Eq. (11)) we obtain 8 ⟨φi1φi2|Hint|φi3φi4⟩= ∞ X k=0 k X mk=−k (k −|mk|)! (k + |mk|)!(−1)(m1+|m1|+m2+|m2|+m3+|m3|+m4+|m4|)/2 × s (2l1 + 1)(l1 −|m1|)! (l1 + |m1|)! s (2l2 + 1)(l2 −|m2|)! (l2 + |m2|)! s (2l3 + 1)(l3 −|m3|)! (l3 + |m3|)! s (2l4 + 1)(l4 −|m4|)! (l4 + |m4|)! × Z ∞ 0 Z ∞ 0 Rn1,l1(r1)Rn2,l2(r2)Rn3,l3(r1)Rn4,l4(r2) rk < rk+1 > r2 1r2 2dr1dr2 ×1 2 Z π 0 P |m1| l1 (cosθ1)P |m3| l3 (cosθ1)P |mk| k (cosθ1)sinθ1dθ1 ×1 2 Z π 0 P |m2| l2 (cosθ2)P |m4| l4 (cosθ2)P |mk| k (cosθ2)sinθ2dθ2 × 1 2π Z 2π 0 ei(−m1+m3+mk)φ1dφ1 × 1 2π Z 2π 0 ei(−m2+m4−mk)φ2dφ2. (A4) 2. Simplifications and Selection Rules The last two integrals in Eq. (A4) require mk to be fixed; moreover, compatibility between the two leads to a condition on quantum numbers of the states that lead to a non-zero expectation value of the interaction potential: m1 + m2 = m3 + m4. (A5) Inspection of the states (A1a - A1f) and those beyond shows that many states (e.g. A1c and A1e) do not con- tribute to the ground state, and hence can be discarded from further discussion. Furthermore, Gaunt was able to evaluate the θ1 and θ2 integrals analytically.25 We have actually found it simpler to evaluate the integrals as they are (either numerically or analytically with the aid of Maple or Mathematica); nonetheless Gaunt’s formula leads to24 the so-called tri- angular condition for the angular momenta. In the first (θ1) integration in Eq. (A4), for example, this requires ℓ1 + ℓ3 ≥k ≥|ℓ1 −ℓ3|. (A6) Thus, the k-sum in Eq. (A4) is truncated at kmax, where kmax = min(ℓ1 + ℓ3, ℓ2 + ℓ4). (A7) It is conventional24,26 to introduce coefficients defined as follows: ck(ℓm; ℓ′m′) = (−1)(m+|m|+m′+|m′|+m−m′+|m−m′|)/2 × s (k −|m −m′|)! k + |m −m′|)! s (2ℓ+ 1)(ℓ−|m|)! (ℓ+ |m|)! s (2ℓ′ + 1)(ℓ′ −|m′|)! (ℓ′ + |m′|)! ×1 2 Z +1 −1 P |m| ℓ (µ)P |m′| ℓ′ (µ)P |m−m′| k (µ)dµ. (A8) In terms of these coefficients the interaction matrix element can be written as ⟨φi1φi2|Hint|φi3φi4⟩= e2 4πǫ0 δm1+m2,m3+m4 Q X k=0 ck(ℓ1m1; ℓ3m3)ck(ℓ4m4; ℓ2m2)Rk(n1ℓ1; n2ℓ2; n3ℓ3; n4ℓ4), (A9) where Rk(n1ℓ1; n2ℓ2; n3ℓ3; n4ℓ4) = Z ∞ 0 Z ∞ 0 Rn1,ℓ1(r1)Rn2,ℓ2(r2)Rn3,ℓ3(r1)Rn4,ℓ4(r2) rk < rk+1 > r2 1r2 2dr1dr2 (A10) This last double integration can be readily done by hand (though it is tedious) or can be", + "4πǫ0 δm1+m2,m3+m4 Q X k=0 ck(ℓ1m1; ℓ3m3)ck(ℓ4m4; ℓ2m2)Rk(n1ℓ1; n2ℓ2; n3ℓ3; n4ℓ4), (A9) where Rk(n1ℓ1; n2ℓ2; n3ℓ3; n4ℓ4) = Z ∞ 0 Z ∞ 0 Rn1,ℓ1(r1)Rn2,ℓ2(r2)Rn3,ℓ3(r1)Rn4,ℓ4(r2) rk < rk+1 > r2 1r2 2dr1dr2 (A10) This last double integration can be readily done by hand (though it is tedious) or can be done (symbolically) with Mathematica or Maple, since the radial wave functions can be readily expressed in terms of associated Laguerre 9 polynomials.17 Also note the reversed order of the argu- ments in the second ck in Eq. (A9); this is important since ck(ℓm; ℓ′m′) = (−1)(m′−m)ck(ℓ′m′; ℓm). (A11) Inspection of the states (A1a-A1f) and those beyond not already omitted by the condition (A5) indicates that three distinct possibilities remain (e.g. when (n1ℓ1m1) = (n2ℓ2m2) and (n3ℓ3m3) = (n4ℓ4m4), etc.) but all of these can be handled through Eq. (A9). One other selection rule is present in these results, though not readily apparent. In Eq. (A9) the same value of the index k must work for both ck coefficients. These coefficients have been tabulated, for example, in the texts by Slater,24 where it is clear that two even or two odd angular momenta (referring to ℓand ℓ′) couple to one another only through even values of k, while an even and an odd angular momentum couple to one another only through an odd value of k. For example ℓ= 0 and ℓ′ = 0 (ss) results in a non-zero ck coefficient only if k = 0, while ℓ= 0 and ℓ′ = 2 (sd) yields a non-zero ck coeffi- cient only if k = 2; the case ℓ= 1 and ℓ′ = 1 (pp) has a non-zero ck coefficient if k = 0 or k = 2. In contrast the sp, sf, and pd coefficients are non-zero for k = 1, k = 3, and k = 1, 3, respectively. This means that ℓ1 +ℓ3 has to have the same parity as ℓ2 + ℓ4. Since the parity of the 1s ↑1s ↓state is even, then, for example, we can further discard state (A1d) from the list, leaving only 3 of the original 6 states listed. 3. An example By way of example we quote some steps for the first state (not listed) that does not utilize single particle s states, ψ14(⃗r1, ⃗r2) = 1 √ 2 \u0002 φ21−1(⃗r1)φ211(⃗r2)+φ21−1(⃗r2)φ211(⃗r1) \u0003 . (A12) To compute the matrix element that couples the first (and primary) basis state with this one, we need to eval- uate the overlap integral, H1,14 = ⟨ψ1|Hint|ψ14⟩. (A13) This consists of two overlaps (Eq. (A12) has two terms), but, through a change of variables ⃗r1 ↔⃗r2, these are readily seen to equal one another. We are thus left with H1,14 = √ 2⟨φ100φ100|Hint|φ21−1φ211⟩. (A14) Eq. (A6) tells us that 1 ≥k ≥1, i.e. only k = 1 need be considered. Then a straightforward evaluation of c1(00; 1 −1) and c1(11; 00) (as required in Eq. (A9)) gives us −1/ √ 3 for the first and +1/ √ 3 for the second (using Eq. (A11)). Next only one double radial integral (Eq.", + "≥1, i.e. only k = 1 need be considered. Then a straightforward evaluation of c1(00; 1 −1) and c1(11; 00) (as required in Eq. (A9)) gives us −1/ √ 3 for the first and +1/ √ 3 for the second (using Eq. (A11)). Next only one double radial integral (Eq. (A10) for k = 1) is required, and a straightforward evaluation, readily done by hand, gives R1(10; 10; 21; 21) = 112 2187 Z a0 . (A15) Combining this with the c1’s and the √ 2 from Eq. (A14) gives H1,14 = − √ 2 448 6561E0, (A16) where E0 ≡ ¯h2 2ma2 0 ≈13.606 eV is adopted as our unit of energy, and we have used a0 = 4πǫ0 e2 ¯h2 m . In practice we have written a program in Maple to per- form these tasks, for the bound basis states, up to some cutoff. That is, all the matrix elements, Hij, up to some cutoff, imax, are evaluated, and then this matrix is diag- onalized. The cutoffimax is defined as the ‘n’ quantum number up to which all states are included. For example, if imax = 2, then only 5 basis states are considered. The rest of the states either do not contribute to the ground state, or, if one of the Hydrogenic single particle states has n = 3, only contributes to the next shell (imax = 3) and beyond. Matrix elements for the case imax = 2 are tabulated in the next section. 4. Matrix equation for Imax = 2 If we restrict basis states in Section II to those with n = 2 or less, only 5 two particle states need to be considered; referring to Eqs. (A1a-A1f), these are ψ1, ψ2, ψ6, ψ14, and ψ16, where ψ14 is given in Eq. (A12), and ψ16(⃗r1, ⃗r2) = φ210(⃗r1)φ210(⃗r2). (A17) Writing the wave function in terms of these wave func- tions alone gives rise to a 5 × 5 matrix diagonalization problem (see Eq. (7)). The resulting equation is:   11 2 −32768 64827 −64 729 −448 6561 √ 2 −448 6561 −32768 64827 2969 729 −4096 84375 2048 28125 √ 2 2048 28125 −64 729 −4096 84375 179 128 −15 128 √ 2 −15 128 −448 6561 √ 2 2048 28125 √ 2 −15 128 √ 2 47 40 −27 640 √ 2 −448 6561 2048 28125 −15 128 −27 640 √ 2 779 640     a1 a2 a7 a14 a16   = E E0   a1 a2 a7 a14 a16   (A18) 10 where E0 ≈13.606 and the subscripts on the coefficients correspond to the labels in the wave functions. The re- sulting ground state energy is E1 = −77.13 eV, and the eigenvector has components a100,100 = 0.9520 a100,200 = −0.3040 a200,200 = −0.0146 a21−1,211 = 0.0267 a210,210 = −0.0188. (A19) Clearly the ψ(100,100) basis state dominates, and small ad- justments occur as the number of basis states increases. Nonetheless, almost 10% of the wave function is com- prised of components beyond the 1s ↑1s ↓state. The", + "0.9520 a100,200 = −0.3040 a200,200 = −0.0146 a21−1,211 = 0.0267 a210,210 = −0.0188. (A19) Clearly the ψ(100,100) basis state dominates, and small ad- justments occur as the number of basis states increases. Nonetheless, almost 10% of the wave function is com- prised of components beyond the 1s ↑1s ↓state. The matrix construction and diagonalization indicated here is repeated, with the aid of the software package MAPLE, for increasing values of Imax, and the results from these calculations are reported in the text. Appendix B: Hylleraas wave function 1. The expectation value of the energy For reference, the expectation value of the energy, us- ing the normalized Hylleraas wave function 1 √ N |ΨHy⟩ with |ΨHy⟩given by Eq. (16), is EHy = −¯h2 2ma2 0 4M N (kZ)2, (B1) where k can be determined analytically in terms of c1 and c2 [see Eqs. (B4), below], kZ = ZL −L′ 2M , (B2) and the unit of energy is the Rydberg, ¯h2 2ma2 0 ≡1 Ryd ≈13.606 eV. (B3) The parameters L, L′, M, and N, corresponding to dif- ferent parts of the Hamiltonian, are given by L =4 + 30c1 + 48c2 + 72c2 1 + 280c1c2 + 576c2 2 L′ =5/4 + 8c1 + 9c2 + (35/2)c2 1 + 48c1c2 + 78c2 2 M =2 + (25/2)c1 + 24c2 + 32c2 1 + 146c1c2 + 480c2 2 N =4 + 35c1 + 48c2 + 96c2 1 + 308c1c2 + 576c2 2. (B4) Aside from units (most treatments use so-called chem- istry units) these all agree with results in the literature.3,20 Use of the optimally determined param- eters, c1 ≈0.0803 and c2 ≈0.0099, and k ≈0.908 deter- mines the energy via Eq. (B1). 2. Overlap integrals: bound states With a very accurate wave function in hand, we can simply utilize Eq. (13) to determine the probability of each basis state in the ground state wave function. The general Hydrogenic bound state can be written as φn,l,m(r, θ, φ) = Gn,ℓe−Zr/na0rℓL2ℓ+1 n−ℓ−1 \u00122Zr na0 \u0013 Y m ℓ(θ, φ), (B5) where Gn,l = s\u0012 2Z na0 \u00133 (n −ℓ−1)! 2n[(n + ℓ)!]3 \u0012 2Z na0 \u0013ℓ , (B6) and the required overlap integrals consist only of terms of the form I = Z φ∗ n1ℓ1m1φ∗ n2ℓ2m2 1 √ N ΨHy. (B7) In the u, s, t coordinates defined above, and using the volume element, dτ = a6 0 8 (s2 −t2)u dsdtdu sin θ1dθ1dφ1dφ2, (B8) many of these integrals are straightforward to calculate analytically — an example will be shown below. How- ever, it is of interest to compute these overlaps for gen- eral quantum numbers, in particular to check on selec- tion rules, and to examine the convergent behaviour for large n1 and n2. To this end we use the same expansion, Eq. (A2) used earlier to evaluate matrix elements of the Hamiltonian, and then evaluate Eq. (B7). The details of the derivation for a general matrix element are a little too cumbersome to include here; all of the overlaps involv- ing bound state wave functions can be expressed in terms of Hypergeometric", + "(A2) used earlier to evaluate matrix elements of the Hamiltonian, and then evaluate Eq. (B7). The details of the derivation for a general matrix element are a little too cumbersome to include here; all of the overlaps involv- ing bound state wave functions can be expressed in terms of Hypergeometric functions, which are readily evaluated using Mathematica, and with these we can readily sum the contributions from the bound states. For students, however, it is more instructive to com- pute this overlap ‘by hand’ for a few of the most relevant states. We outline the procedure for the most important state, ψ1(⃗r1, ⃗r2) = φ100(⃗r1)φ100(⃗r2). Then a1 = Z d3r1 Z d3r2 φ100(⃗r1)φ100(⃗r2) 1 √ N ΨHy(⃗r1, ⃗r2), (B9) and the only angular dependence occurs in the |⃗r1 −⃗r2| term in the Hylleraas wave function. This requires knowl- edge of the angle between the two vectors, ⃗r1 and ⃗r2, which we can call θ12; this is given as3 cos θ12 = cos θ1 cos θ2+sin θ1 sin θ2 cos (φ2 −φ1), (B10) where ⃗ri ≡(ri, θi, φi), for i = 1, 2 are the spherical coor- dinates for each vector. A simpler trick is to line up the 11 z2 axis with ⃗r1; then θ12 ≡θ2, and all six integrations are then straightforward. The result is a1 = 32 √ N k3 (k + 1)6 \u001a 4 + 35c1 k k + 1 + 96c2 \u0000k k + 1 \u00012 \u001b , ≈0.9624, (B11) and therefore a2 1 ≈0.9262. The numerical values follow upon substitution of the optimal values of k, c1, and c2. Similar calculations can be performed for the other over- lap coefficients; some results are tabulated in Table 1, where the constituents of the basis states, suitably sym- metrized, are listed. Note that the label provided under the ‘basis state’ column identifies the two single particle wave functions involved, and the basis state is a singlet state, and therefore the spatial part is symmetrized, as enumerated in Eqs. (A1a-A1f). Appendix C: Evaluation of overlap integrals We provide some of the details for the evaluation of Eq. (21). Because of the symmetry in the singlet state , and using the definition ap = r 2 N 2 π \u0000kZ a0 \u00013Ap (C1) we require the integral Ap = Z d3r1 Z d3r2φp(r1)φ100(r2)e−Zk a0 (r1+r2)\b 1 + 2Zc1 k a0 |⃗r1 −⃗r2| + c2 \u0002 2Z k a0 (r1 −r2) \u00032 , (C2) where φp(r1) = 1 √ 4π Z a0 s 2π pa0 Z 1 −e−2πZ/(pa0) e−iprM(1 + i Z pa0 , 2, 2ipr), (C3) and M(a,b,z) is the Kummer function.21 Note that Z π 0 dθ1 sin θ1 Z π 0 dθ2 sin θ2 Z 2π 0 dφ1 Z 2π 0 dφ2 [1] = (4π)2, (C4) while Z π 0 dθ1 sin θ1 Z π 0 dθ2 sin θ2 Z 2π 0 dφ1 Z 2π 0 dφ2|⃗r1 −⃗r2| = (4π)2\u0000r> + 1 3 r2 < r> \u0001 , (C5) where r< (r>) refers the smaller (larger) of r1 or r2. The first of these is trivial, while the second", + "0 dθ1 sin θ1 Z π 0 dθ2 sin θ2 Z 2π 0 dφ1 Z 2π 0 dφ2|⃗r1 −⃗r2| = (4π)2\u0000r> + 1 3 r2 < r> \u0001 , (C5) where r< (r>) refers the smaller (larger) of r1 or r2. The first of these is trivial, while the second follows most read- ily by using the trick mentioned following Eq. (B10). Two integrations remain, and it is efficient to switch to dimensionless variables, x1 ≡Zr1/a0, x2 ≡Zr2/a0, and y ≡pa0/Z. Then, since the Kummer function depends only on x1, the x2 integration is elementary, and leaves behind polynomials in x1. One arrives at the expression given by Eq. (22), with the definitions In(y) ≡Bn(y, k + iy) and Jn(y) ≡Bn(y, 2k + 1 + iy), (C6) where Bn(y, z) ≡ Z ∞ 0 dx xne−zxM(1 + i y , 2, 2iyx). (C7) Now an expansion of the Kummer function (following Eq. (19)) allows us to do the integral and obtain an infinite summation which can be recognized as a Hypergeometric function,21 Bn(y, z) = n! zn+1 F[1 + i y , n + 1; 2; 2iy z ], (C8) with n = 1, 2, 3, 4 needed. The standard definition of the Hypergeometric function uses the Gauss hypergeometric series,21 with circle of convergence in the unit circle |z| = 1 F(a, b; c; z) = ∞ X m=0 (a)m(b)m (c)m zm m! , (C9) where the (a)m are the Pochhammer symbols introduced earlier. Writing Eq. (C8) wouldn’t normally be too use- ful (because Hypergeometric functions are hard to eval- uate numerically), except that one can use the identity21 F[a, b; c; z] ≡(1 −z)c−a−bF[c −a, c −b; c; z]. (C10) This identity is extremely helpful because the second variable in the Hypergeometric function on the right 12 hand side is a non-positive integer for n = 1, 2, 3, 4, and so, because of the definition of the Pochhammer symbol, the infinite sum in the definition of the Hypergeometric function becomes truncated to n terms. We thus obtain, for n = 1, 2, 3, 4, In(y) = 1 k2 + y2 exp \u0000−2 ytan−1 y k \u0001 Dn (C11) and Jn(y) = 1 (2k + 1)2 + y2 exp \u0000−2 ytan−1( y 2k + 1) \u0001 En, (C12) where Dn = n! (k −iy)n−1 F[1 −i y , 1 −n; 2; 2iy k + iy], (C13) and En = n! (2k + 1 −iy)n−1 F[1 −i y , 1 −n; 2; 2iy 2k + 1 + iy]. (C14) With these definitions, straightforward evaluation gives D1 = 1 D2 = 2(k −1) k2 + y2 D3 = 4(k −1)(2k −1) (k2 + y2)2 − 2 k2 + y2 D4 = 8(k −1)(2k −1)(3k −1) (k2 + y2)3 −8(3k −2) (k2 + y2)2 (C15) and E1 = 1 E2 = 4k (2k + 1)2 + y2 . (C16) Summarizing, we have I1 = 1 k2 + y2 exp \u0000−2 ytan−1 y k \u0001 I2 = 2(k −1) (k2 + y2)2 exp \u0000−2 ytan−1 y k \u0001 I3 = \u001a4(k −1)(2k −1) (k2 +", + "(C15) and E1 = 1 E2 = 4k (2k + 1)2 + y2 . (C16) Summarizing, we have I1 = 1 k2 + y2 exp \u0000−2 ytan−1 y k \u0001 I2 = 2(k −1) (k2 + y2)2 exp \u0000−2 ytan−1 y k \u0001 I3 = \u001a4(k −1)(2k −1) (k2 + y2)3 − 2 (k2 + y2)2 \u001b exp \u0000−2 ytan−1 y k \u0001 I4 = \u001a8(k −1)(2k −1)(3k −1) (k2 + y2)4 −8(3k −2) (k2 + y2)3 \u001b exp \u0000−2 ytan−1 y k \u0001 J1 = 1 (2k + 1)2 + y2 exp \u0000−2 ytan−1( y 2k + 1) \u0001 J2 = 4k ((2k + 1)2 + y2)2 exp \u0000−2 ytan−1( y 2k + 1) \u0001 (C17) and these are to be substituted into Eq.(22). 1 E.A. Hylleraas, “On the Ground State of the Helium Atom,” Z. Phys. 48, 469 (1929). The work in this paper by Dr. Biem¨uller was never published, to our knowledge. Hylleraas’ paper and many others are reprinted in English in Ref. [2]. 2 H. Hettema, Quantum Chemistry: Classic Scientific Pa- pers (World Scientific, Singapore, 2000). 3 E.A. Hylleraas,“A New Calculation of the Energy of He- lium in the Ground State as Well as the Lowest Term of Ortho-helium, Z. Phys. 54, 347 (1929). This is also reprinted in English in Ref. [2]. See also Mathematical and Theoretical Physics, Volume II, by E.A. Hylleraas (Wiley- Interscience, Toronto, 1970), particularly Chapter 7. 4 We use the phrase, “singlet state 1s ↑1s ↓” to refer to the spin wave function 1 √ 2 \u0002 1s ↑1s ↓−1s ↓1s ↑ \u0003 . 5 That a 1s ↑1s ↓-like state (i.e. a non-Hydrogenic S state with zero total angular momentum) will ultimately suc- ceed was recognized early by Eugene Wigner, as recorded in E.A. Hylleraas, “Reminisences of Early Quantum Me- chanics of Two-Electron Atoms,” Rev. Mod. Phys 35, 421 (1963). 6 See, for example, C. Schwartz, “Experiment and Theory in Computations of the He Atom Ground State,” Inter- national Journal of Modern Physics E 15, 877 (2006); in the update (arXiv:math-ph/0605018v1), the ground state energy is determined with an accuracy of 50 digits. 7 We should note that this work was started without knowl- edge of Hylleraas’ remark concerning Dr. Biem¨uller quoted in the opening paragraph, and we initially expected that convergence would be attained with inclusion of a few shells beyond the singlet state 1s ↑1s ↓’ configuration. 8 Again, for those that have reflected on this problem, the need to include continuum (or scattering) states comes as no surprise, and they may even be inclined to dismiss the entire exercise as ‘bad results due to starting with a poor basis set.’ But we re-iterate, the ‘poor basis set’ with which we start is the one with which we tend to think, and there- fore these questions are worth answering. Furthermore, as outlined in the subsequent discussion, in some research- type problems we are forced to adopt a poor basis set, and then the present analysis may provide some useful guid- ance for these problems. 9 J. E. Hirsch, “Dynamic Hubbard", + "think, and there- fore these questions are worth answering. Furthermore, as outlined in the subsequent discussion, in some research- type problems we are forced to adopt a poor basis set, and then the present analysis may provide some useful guid- ance for these problems. 9 J. E. Hirsch, “Dynamic Hubbard Model,” Phys. Rev. Lett. 87, 206402-1-4 (2001); “Quantum Monte Carlo and exact diagonalization study of a dynamic Hubbard model,” Phys. Rev. B 65, 214510-1-16 (2002); “Quasiparticle undress- ing in a dynamic Hubbard model: Exact diagonalization study,” Phys. Rev. B 66, 064507-1-12 (2002). 10 G. H. Bach, J. E. Hirsch and F. Marsiglio, “Two-site dynamical mean field theory for the dynamic Hubbard 13 model,” Phys. Rev. B 82, 155122-1-15 (2010). 11 G. H. Bach and F. Marsiglio, “Optical conductivity for a dimer in the dynamic Hubbard model,” Phys. Rev. B 85, 155134-1-10 (2012). 12 J. Hubbard, “Electron Correlations in Narrow Energy Bands,” Proc. R. Soc. London A 276, 238-257 (1963); “Electron Correlations in Narrow Energy Bands. II. The Degenerate Band Case,” 277, 237-259 (1964); “Electron Correlations in Narrow Energy Bands. III. An Improved Solution.” 281, 401-419 (1964); “Electron Correlations in Narrow Energy Bands. IV. The Atomic Representation,” 285, 542-560 (1965). 13 A search through APS journals alone reveals over 5000 articles that contain the keyword ‘Hubbard’ in their title or abstract. 14 This is reminiscent of the situation in General Relativity, where the presence of mass alters the space-time structure in which the mass is situated. 15 Actually the method used to determine the ‘exact’ ground state energy and wave function (e.g. in Ref. [6]) is a straightforward generalization of the simplest Hylleraas wave function quoted here. 16 Ira N. Levine, Quantum Chemistry, 6th Edition (Pearson, Upper Saddle River, 2009). 17 D. J. Griffiths, Introduction to Quantum Mechanics (Pear- son/Prentice Hall, Upper Saddle River, NJ, 2005), 2nd edition. 18 Ironically, we use particle labels to avoid particle labelling, i.e. to enforce indistinguishability. See Footnote4 for equiv- alent expressions of the spin singlet state. 19 F. Marsiglio, “The harmonic oscillator in quantum me- chanics: a third way,” Am. J. Phys. 77, 253 (2009). 20 E.W. Schmid, G. Spitz, and W. L¨osch, Theoretical Physics on the Personal Computer, (Springer Verlag, New York, 1987), Chap. 16. 21 M. Abramowitz and I.A. Stegun, Handbook of Mathemat- ical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, Inc. New York, 1972). 22 L.D. Landau and E.M. Lifshitz, Quantum Mechanics, 2nd Edition, (Pergamon Press, New York, 1965), Eq. (33.4). 23 J.D. Jackson, Classical Electrodynamics, 2nd Edition (John Wiley and Sons, Toronto, 1975). 24 J.C. Slater, Quantum Theory of Atomic Structure, (McGraw-Hill, Toronto, 1960). The required formula is found on p. 308. 25 J.A. Gaunt, “The Triplets of Helium,” Phil. Trans. Roy. Soc. (London) A228:151 (1929). 26 E.U. Condon and H.H. Shortley, Theory of Atomic Spectra, (Cambridge University Press, New York, 1935).", + "H.H. Shortley, Theory of Atomic Spectra, (Cambridge University Press, New York, 1935).", + "1 Direct demonstration of the emergent magnetism resulting from the multivalence Mn in a LaMnO3 epitaxial thin film system Wei Niu, Wenqing Liu, Min Gu, Yongda Chen, Xiaoqian Zhang, Minhao Zhang, Yequan Chen, Ji Wang, Jun Du, Fengqi Song, Xiaoqing Pan, Nini Pryds, Xuefeng Wang,* Peng Wang,* Yongbing Xu,* Yunzhong Chen,* and Rong Zhang W. Niu, Y. D. Chen, X. Q. Zhang, M. H. Zhang, Y. Q. Chen, Prof. X. F. Wang, Prof. Y. B. Xu and Prof. R. Zhang National Laboratory of Solid State Microstructures and Jiangsu Provincial Key Laboratory of Advanced Photonic and Electronic Materials, School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China E-mail: xfwang@nju.edu.cn; ybxu@nju.edu.cn; W. Niu, Prof. N. Pryds and Dr. Y. Z. Chen Department of Energy Conversion and Storage, Technical University of Denmark, Risø Campus, Roskilde 4000, Denmark E-mail: yunc@dtu.dk Dr. W. Q. Liu Department of Electronic Engineering, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom M. Gu, Prof. P. Wang and Prof. X. Q. Pan College of Engineering and Applied Sciences, Nanjing University, Nanjing 210093, China E-mail: wangpeng@nju.edu.cn J. Wang, Prof. J. Du and Prof. F. Q. Song School of Physics, Nanjing University, Nanjing 210093, China Prof. J. Du, Prof. F. Q. Song, Prof. X. Q. Pan, Prof. X. F. Wang, Prof. P. Wang, Prof. Y. B. Xu, and Prof. R. Zhang Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Prof. X. Q. Pan Department of Chemical Engineering and Materials Science, University of California Irvine, Irvine, CA 92697, USA 2 Abstract: Atomically engineered oxide heterostructures provide a fertile ground for creating novel states. For example, a two-dimensional electron gas at the interface between two oxide insulators, giant thermoelectric Seebeck coefficient, emergent ferromagnetism from otherwise nonmagnetic components, and colossal ionic conductivity. Extensive research efforts reveal that oxygen deficiency or lattice strain play an important role in determining these unexpected properties. Herein, by studying the abrupt presence of robust ferromagnetism (up to 1.5 B/Mn) in LaMnO3-based heterostructures, we find the multivalence states of Mn that play a decisive role in the emergence of ferromagnetism in the otherwise antiferromagnetic LaMnO3 thin films. Combining spatially resolved electron energy-loss spectroscopy, X-ray absorption spectroscopy and X-ray magnetic circular dichroism techniques, we determine unambiguously that the ferromagnetism results from a conventional Mn3+-O-Mn4+ double-exchange mechanism rather than an interfacial effect. In contrast, the magnetic dead layer of 5 unit cell in proximity to the interface is found to be accompanied with the accumulation of Mn2+ induced by electronic reconstruction. These findings provide a hitherto-unexplored multivalence state of Mn on the emergent magnetism in undoped manganite epitaxial thin films, such as LaMnO3 and BiMnO3, and shed new light on all-oxide spintronic devices. 3 1. Introduction Complex oxides provide a rich platform for creating novel states and functional properties, especially at the interface of heterostructures and superlattices.[1] The interplay between charge, spin, orbital and lattice degrees of freedom gives rise to a rich spectrum of fascinating phenomena at complex oxide interfaces, including high-mobility two-dimensional electron gases,[2,3] giant thermoelectric Seebeck coefficient,[4] multiferroic properties,[5] colossal ionic conductivity[6] and intriguing", + "functional properties, especially at the interface of heterostructures and superlattices.[1] The interplay between charge, spin, orbital and lattice degrees of freedom gives rise to a rich spectrum of fascinating phenomena at complex oxide interfaces, including high-mobility two-dimensional electron gases,[2,3] giant thermoelectric Seebeck coefficient,[4] multiferroic properties,[5] colossal ionic conductivity[6] and intriguing magnetic properties.[7,8] Often, oxygen deficiency and/or strain effect play a key role in determining these unexpected properties.[6,9-12] For example, unintentional introduction of oxygen vacancies on the SrTiO3 (STO) side contributes significantly to the metallic conduction at the interface of LaAlO3/SrTiO3 (LAO/STO).[13,14] The effect of oxygen excess at complex oxide interfaces has long been neglected since atomically controlled fabrication of oxide heterostructures are mostly achievable under oxygen deficient conditions. Consequently, to date, a comprehensive study to understand it and its role in the novel properties of the strongly correlated complex oxides has not been fully reached. The stoichiometric bulk-state LaMnO3 (LMO) is an A-type antiferromagnetic insulator with the orthorhombic perovskite crystal structure.[15,16] However, it turns to be a ferromagnetic insulator when formed as thin films.[17-19] The origin of such emergent ferromagnetic state has been vigorously investigated but the underlying mechanisms remains controversial.[17,18,20-22] Using scanning superconducting quantum interference device (SQUID) microscopy, ferromagnetism in LMO/STO heterostructure was attributed to stem from an electronic reconstruction at the interface due to the polar discontinuity.[17,18,23] However, such polarity-related electronic reconstruction normally occurs in proximity to the sample surface or interface. This can hardly explain the ferromagnetism which only occurs when the LMO film is thick. Besides, spin and orbital Ti magnetism was suspected to be 4 responsible for the ferromagnetic behaviors in the LMO/STO superlattices.[21,24] Nevertheless, since most of the Mn eg bands locate approximately 1 eV below the conduction band of STO (Ti 3d t2g), the Ti magnetism is expected to be quite small.[25,26] Moreover, the epitaxial strain induced by the lattice mismatch between the film and substrate has also been suggested an origins for the ferromagnetic state.[27,28] But, as we will see later, this can hardly explain that ferromagnetism is universally observed in LMO films grown on different substrates. In contrast, our demonstration of the multivalence state of Mn can explain both the previously reported data and the new results present here. Notably, oxygen excess is often induced by the cation vacancies[29-32] since the perovskite structure cannot accept the excessive oxygen in an interstitial site. Although oxygen excess has been put forward as a possible explanation for the ferromagnetism in very thick LMO films (100-150 nm),[33] it remains open whether such ferromagnetism can persist in proximity to the interface. In this article, we report that the antiferromagnetic LMO can easily adopt the excessive oxygen from its stoichiometric phase, thus the multivalence states of Mn appear and show emergent ferromagnetism when epitaxially grown in thin film form via a well-controlled layer-by-layer two-dimensional growth mode. The ferromagnetism shows up abruptly when the film thickness is above 6 unit cell (uc). It is sensitive to the oxygen deposition pressure (PO2) but insensitive to the type of the substrate (i.e., the strain). By combing spatially resolved electron energy-loss spectroscopy (EELS),", + "via a well-controlled layer-by-layer two-dimensional growth mode. The ferromagnetism shows up abruptly when the film thickness is above 6 unit cell (uc). It is sensitive to the oxygen deposition pressure (PO2) but insensitive to the type of the substrate (i.e., the strain). By combing spatially resolved electron energy-loss spectroscopy (EELS), element-specific X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD), we find that the mixed valence state of Mn ions (Mn4+, Mn3+ and Mn2+) exists and the ferromagnetism is attributed to the Mn3+-O-Mn4+ double-exchange mechanism. In addition, the magnetic dead-layer effect, i.e., the strong depression of magnetic properties when the thickness of the film is below 6 uc, is found to be associated with an accumulation of Mn2+ induced by electronic reconstruction in the proximity of the interface. 5 2. Result and discussion 2.1. Emergent ferromagnetism in LMO-based heterostructures LMO (001) thin films with thickness ranging from 3 to 20 uc were grown by pulsed laser deposition (PLD) on different perovskite substrates of STO (001), (LaAlO3)0.3(Sr2AlTaO6)0.7 (LSAT) (001) and LaAlO3 (LAO) (001), as schematically shown in Figure 1a. The film growth is in-situ monitored by the reflection high-energy electron diffraction (RHEED) under different PO2 (10-7-10-3 mbar), where layer-by-layer film growth is achieved (Figure 1b). The sharp streaky line of the RHEED patterns in Figure 1c, d and e after deposition further indicate high quality of the films. All films show atomically smooth surface as confirmed by atomic force microscopy (Figure 1f). Note that, previous reports indicate that the as-deposited LMO thin films already show noticeable oxygen excess at a deposition pressure of 10-3 mbar.[33, 34] Figure 2a shows the temperature-dependent magnetic moment (M-T) of LMO/STO heterostructures with different film thickness (t). When t ≤ 5 uc, no ferromagnetism is observed. In contrast, at t = 6 uc, the LMO film abruptly turns ferromagnetic with Curie temperature (TC) of ~95 K and a saturation magnetization of ~ 0.5 B/Mn (at B =0.1 T). Later on, the ferromagnetism develops gradually with the increase in the film thickness. Figure 2b summarizes the LMO saturation magnetic moment (Ms) as a function of thickness measured at 10 K. After the sharp onset of ferromagnetic order at t = 6 uc, the Ms of the LMO/STO heterostructures increases with the increasing t. When t ≥ 10 uc, the Ms saturates at a value of ~1.5 B/Mn. This behavior indicates that the measured magnetism comes from the majority of the “bulk” of the film rather than an interfacial effect. If it were, Ms would have shown a decrease as the thickness of the LMO layer increases.[21] We further find that the highly ferromagnetic state is not limited to the LMO/STO heterostructures (the lattice parameter of STO is 3.91 Å), but can be universally observed 6 even when LMO film is epitaxially grown on two other substrates with different lattice parameters and polarities, i.e. LSAT (a=3.86 Å) and LAO (a=3.79 Å). For comparison, Figure 2c shows the M-T curves for LMO films (20 uc) grown on STO, LSAT and LAO substrates, respectively. All these LMO-based heterostructures evidently demonstrate the", + "film is epitaxially grown on two other substrates with different lattice parameters and polarities, i.e. LSAT (a=3.86 Å) and LAO (a=3.79 Å). For comparison, Figure 2c shows the M-T curves for LMO films (20 uc) grown on STO, LSAT and LAO substrates, respectively. All these LMO-based heterostructures evidently demonstrate the emergent ferromagnetism despite the difference of Ms. The Ms of heterostructures of LMO/STO, LMO/LSAT and LMO/LAO is found to be 1.56 B/Mn, 0.84 B/Mn and 0.72 B/Mn, respectively. The inset of Figure 2c displays the field-dependent in-plane magnetization (T = 10 K) of these three heterostructures with dominant hysteresis loops, further suggesting the robust ferromagnetism. Since these epitaxial films exhibit different content of coherent strains,[18] the presence of ferromagnetism in these three different types of heterostructures clearly rules out the strain effect as the dominant origin for the occurrence of ferromagnetism. To explore the influence of oxygen pressure on the magnetism, LMO films were further grown under different PO2 conditions. Figure 2d shows a comparison of M-T curves of the LMO films (t =10 uc) grown at different PO2. With the PO2 increasing from 10-7 mbar to 10-3 mbar, the magnetization is enhanced by nearly 6 times in magnitude up to a value of ~1.3 B/Mn. Moreover, the TC is also significantly increased from ~50 K to ~137 K upon the increase in PO2 (inset of Figure 2d). Therefore, it is clear from these results that the ferromagnetism of LMO-based heterostructures shows strong dependence on the oxygen atmosphere during the deposition. 2.2. Multivalence state of Mn ions—STEM-EELS observation Figures 3a and b show chemically atomic-resolution high-angle annular dark field (HAADF) scanning transmission electron microscope (STEM) images of the cross-section 10-uc LMO/STO heterostructure of samples grown at 1×10-3 mbar and 1×10-7 mbar, respectively. Both samples show a nice continuity of the perovskite-structure stacking 7 sequence across the interface. The interfacial layers are well crystallized without appreciable dislocations or other defects. This confirms the high-quality coherent layer-by-layer epitaxial growth. Notably, these two LMO samples exhibit exactly the same lattice parameters (a=b=~3.884 Å, c=~3.914 Å), i.e. the same interfacial strain state, but they show the different magnetic properties. This further excludes the possibility of lattice strain as a dominant origin of the underlying ferromagnetism. To reveal the intrinsic difference between these two samples, spatially resolved EELS[35] across the LMO/STO interface and on the LMO side is further measured. Figure 3d and e show the Mn-L2,3 edge EELS mapping of the selected area in Figure 3a and b (red rectangles). It can be clearly seen that the Mn-L2,3 edge appears two peaks, corresponding to the excitations from the spin-orbital splitting of 2p3/2 and 2p1/2 levels to empty states in the 3d band.[19] Figure 3f shows the corresponding EELS profile of the ferromagnetic LMO sample (grown at PO2=1×10-3 mbar) from the LMO/STO interface to the LMO film surface. Notably, Mn-L3 edge spectra close to the interface exhibit a slight shift towards the lower energy loss in comparison to those from the surface of the LMO films, which suggests that the chemical valence state of Mn near the", + "PO2=1×10-3 mbar) from the LMO/STO interface to the LMO film surface. Notably, Mn-L3 edge spectra close to the interface exhibit a slight shift towards the lower energy loss in comparison to those from the surface of the LMO films, which suggests that the chemical valence state of Mn near the interface is reduced. A fit to the Mn-L3 edge reveals the presence of multivalence state of Mn2+, Mn3+ and Mn4+ (see Supporting Information Figure S3). The fitting results reveal that a fraction of Mn2+ dominates at the interface (~42%) and then drops to a constant ratio (~20%) at 5 uc far away from the interface. However, the proportion of Mn3+/Mn4+ increases firstly to a maximum at 5 uc and then remains unchanged along with the increased thickness of LMO from the interface to the surface. These findings are further confirmed by the following XAS measurements. Notably, this trend of a reduction of Mn oxidation valence state is similar to the case of optimally doped manganite, such as the ferromagnetic La0.7Sr0.3MnO3/STO[36-39] and the ferromagnetic La0.7Ca0.3MnO3/STO.[40] In these doped manganite films, the ferromagnetism results from the Mn3+-O-Mn4+ double- exchange interaction. The reduced oxidation state (Mn2+) in the sample corresponds to the 8 decrease in the magnetization and has been proposed to explain the observed magnetic dead layer in the most manganite thin films.[38] Figure 3c shows the comparison of O-K edges of two samples. The relative strength of the first peak in green shaded area of O-K edge is an indication of d-band filling.[41] Remarkably, the intensity of the first peak in O-K spectra of LMO film grown at 1×10-7 mbar is lower than the one grown at 1×10-3 mbar. This reduction in the first-peak intensity of O-K edge indicates that more electrons fill in the d-band. Therefore, the weak ferromagnetic LMO film grown at 1×10-7 mbar has a lower valence state of Mn as compared to the robust ferromagnetic LMO film grown under 1×10-3 mbar. The chemical valence state of Mn in the sample grown at 1×10-7 mbar is dominated by Mn3+ with a lower ratio of the Mn4+. As shown in Figure 3g, we compare the Mn-L2,3 edge spectra of 10-uc LMO grown under the different PO2. It is obvious that there is a peak shift towards the higher energy loss for the LMO film grown under 1×10-3 mbar, further proving evidence of the higher content of Mn4+ under the high-oxygen-pressure condition. More quantitative analysis reveals that, compared with the Mn4+ ratio of ~24.6% in the 1×10-3 mbar sample, the ratio of Mn4+ decreases to ~15.9% in the 1×10-7 mbar sample, while the proportion of Mn3+ remains nearly unchanged. In addition, the portion of Mn2+ increases from 19.6% to 24.5% as the PO2 decreases. From these results, we draw the conclusion that the main difference between these two LMO films is that, the ferromagnetic LMO film grown under higher PO2 has more Mn4+ ions and thus a high Mn4+/Mn3+ ratio, which enhances the double-exchange interaction between Mn3+ and Mn4+. 2.3. Multivalence state of Mn ions—XAS/XMCD observation To further unveil the origin of", + "the main difference between these two LMO films is that, the ferromagnetic LMO film grown under higher PO2 has more Mn4+ ions and thus a high Mn4+/Mn3+ ratio, which enhances the double-exchange interaction between Mn3+ and Mn4+. 2.3. Multivalence state of Mn ions—XAS/XMCD observation To further unveil the origin of the emergent ferromagnetism, the element-specific XAS and XMCD at the Mn and Ti L2,3 absorption edges are performed to probe the local electronic character of the magnetic ground state of the LMO. As schematically shown in Figure 4a, circularly polarized X-rays with ~100% degree of polarization were performed at 60o with 9 respect to the film plane and in parallel with the applied magnetic field. Typical XAS spectra of the LMO/STO heterostructures are displayed in Figure 4b. Mn-L2,3 edge XAS spectra show the prominent multiplet structure for both spin-orbit split core levels, indicating a mixture valence state of Mn. For comparison, spatially resolved XAS spectra and the corresponding atomic multiplet calculation of Mn2+, Mn3+ and Mn4+ from Ref. [42] are taken as a reference (see the blue curves in Figure 4b). Remarkably, the experimental spectra are comparable to those reference spectra, showing that the experimental spectra are composed of the Mn2+, Mn3+ and Mn4+ in all LMO films, in good agreement with our EELS results. L3 peak locates at ~642.0 eV, indicating the main valence state of Mn3+. While the stoichiometric LMO crystal contains only Mn3+ ions, oxygen excess introduces Mn4+ ions into the Mn sublattice, resulting in the observed ferromagnetic coupling between the local spins via double-exchange mechanism.[19,43] The robust ferromagnetism is favored in a more oxidizing atmosphere because of the formation of Mn4+ and the resultant double-exchange mechanism relevant to Mn3+-O-Mn4+. The fingerprint of Mn2+ valence state at the lower energies of ~640.1 eV dominates when t < 6 uc, while the Mn3+ and Mn4+ content increase with the increasing t as evidenced by the increase in spectral weight at the photon energy of 642.0 eV and 643.4 eV. Note that the increased Mn4+/Mn3+ ratio with the thickness can rigorously increase the strength of the double-exchange interaction, thereby enhancing the ferromagnetism. Moreover, the spectra exhibit the identical line-shapes, indicating that LMO has the same La/Mn ratio, otherwise a different ratio should have resulted in a different line-shape.[44] This is also consistent with the bulk LMO case. Although an oxygen excess can be obtained, the La/Mn ratio always keeps 1.[31] Ti-L2,3 isotropic XAS measurements for heterostructures with different LMO thickness are shown in Figure 4c. All measurements display a similar electronic structure. The energy difference between the two main peaks of the L3 and L2 edges remain unchanged, as expected for Ti4+ of the STO layer. This suggests no charge transfer to the Ti site. And the charge 10 reconstruction occurs mainly on the Mn site at the interface of LMO/STO. The presence of Ti4+ rather than Ti3+ is also proved independently by our EELS spectra at Ti L-edge (Figure S3), consistent with the previous results,[37] ruling out the possible mechanism of Ti interfacial magnetism.[21,24] In the light of the valence", + "mainly on the Mn site at the interface of LMO/STO. The presence of Ti4+ rather than Ti3+ is also proved independently by our EELS spectra at Ti L-edge (Figure S3), consistent with the previous results,[37] ruling out the possible mechanism of Ti interfacial magnetism.[21,24] In the light of the valence state of Mn ions, we now turn to its contribution to magnetic properties. Representative XAS/XMCD and their integration taken at the Mn-L2,3 edges are shown in Figure 5a. The magnitude of the magnetization is quantitatively estimated using the XMCD spin sum-rules.[45] Within the limit of the uncertainties in the sum-rules estimation, the obtained total magnetization is approximately ~1.5 B/Mn, which is in reasonable agreement with our SQUID results. In addition, the XMCD signal of ~24% at the Mn-L3 edge of 15-uc LMO is consistent with previously measured values.[45] We summarize the thickness dependence of the XMCD-derived magnetic moment in Figure 5b, together with the thickness-dependent magnetization of LMO/STO measured by SQUID. Good agreement of the magnetization derived from both measurements show the same thickness dependence. These results indicate that the ferromagnetism of LMO/STO heterostructures comes from the intrinsic contribution from the exchange interaction of the multivalence Mn ions rather than the interface-driven effect from the electronic reconstruction or the strain-induced effect by the substrates. 2.4. Electronic reconstruction and dead-layer behavior in manganite films For bulk LMO, oxygen excess is usually accommodated by the formation of cation vacancies on both La and Mn sites,[31,46] leaving a perfect oxygen sublattice. To keep neutrality of the charge, a fraction of Mn3+ in the stoichiometric LMO must be oxidized to Mn4+. This is a direct reason that the mixed valence states of Mn (Mn3+ and Mn4+) should be shown in the EELS and XAS spectra. The oxygen excess harasses ferromagnetic Mn3+-O- Mn4+ double-exchange interactions by the Mn4+/Mn3+ ratio.[31,32] The ferromagnetic BiMnO3 11 holds the similar mechanism to LMO due to the oxygen excess.[47] Based on our comparison of O-K and Mn-L edges of LMO deposited at the different oxygen pressure (Figure 3), when the LMO film is grown at PO2=1×10-7 mbar the ratio of Mn4+/Mn3+ is reduced, thereby the double-exchange interaction is weakened and the ferromagnetism is depressed.[48] In contrast, when deposited at a typical oxygen pressure of 10-3 mbar, the higher the Mn4+/Mn3+ ratio the more significant is the double-exchange interaction as well as the robust ferromagnetism. Additionally, LMO is polar whereas STO is nonpolar. Therefore, polar-discontinuity- induced electronic reconstruction could occur at the LMO/STO interface.[2] However, different from the intensively investigated LAO/STO system where the reconstructed electrons are transferred from the sample surface to the STO inside, in LMO/STO system, the empty or partially filled eg bands of LMO are often lower than the Ti 3d bands of STO,[49] as schematically illustrated in Figure 6a. In this vein, the electronic reconstruction in LMO/STO occurs on the LMO side, i.e. from LMO surface to the LMO in proximity to the interface rather than to the STO side.[49-51] This scenario is consistent with the fact that Mn2+ becomes detectable at the interface while no signal of", + "6a. In this vein, the electronic reconstruction in LMO/STO occurs on the LMO side, i.e. from LMO surface to the LMO in proximity to the interface rather than to the STO side.[49-51] This scenario is consistent with the fact that Mn2+ becomes detectable at the interface while no signal of Ti3+ is detected, as evidenced in the EELS and XAS spectra of Ti (Figure 4 and Figure S3). It should be noted that,[18] Wang et al. did not observe ferromagnetism when LMO thin films (12 uc) were grown on the polar substrates of LAO and LSAT. They attributed the non-ferromagnetism to the absence of polar discontinuity because both LMO and substrates (LAO and LSAT) are polar.[18] Herein, when we increase the film thickness to 20 uc, all heterostructures of LMO/STO, LMO/LAO and LMO/LSAT show ferromagnetism (Figure 1c). Hence, the electronic reconstruction due to the polar discontinuity cannot account for the ferromagnetism of LMO but coherent with the absence of magnetism at the interface. The sharp transition from ferromagnetism to non-ferromagnetism at t < 6 uc should suggest a magnetic dead layer of LMO in the vicinity of the interface. As shown in Figure 6b, LMO thin films are spatially separated into two stacked sheets. One is dead layer at the 12 bottom without ferromagnetism, the other is a uniformly magnetized layer on the top. The dead layer is widely observed in other ferromagnetic manganite films, such as La1-xSrxMnO3 (x~0.3 and 0.33) and LMO, as summarized in Figure 6c.[52-54] The mechanism underlying this dead-layer effect remains elusive. However, this might be due to the presence of Mn2+,[38] which always shows a critical thickness (tc) of ~2 nm (i.e., 5 uc) when the substrate is STO. The different critical thickness of the dead layer on various substrates could be due to the fact that films on the different substrates possess the different responses, such as the energy cost of creating vacancies and the capacity of adapting chemical stoichiometry during the deposition.[55-57] 3. Conclusion To conclude, we have firstly established the multivalence state of Mn2+, Mn3+ and Mn4+ ions in LMO epitaxial thin films. The ferromagnetism originates from a conventional Mn3+- O-Mn4+ double-exchange mechanism rather than an interface or strain effect. The observed thickness-dependent ferromagnetism is controlled by the magnetic dead-layer effect in manganite thin films, which is accompanied by the accumulation of Mn2+ induced by the electronic reconstruction. Our results not only shed light on the emergent ferromagnetism of the otherwise antiferromagnetic LMO thin films, but also broaden the understanding of oxygen excess in complex oxides, which is beneficial for the rational design of future all- oxide spintronic devices, particularly based on LMO. 4. Experimental Section Epitaxial Growth: The LMO films were grown by PLD technique (KrF excimer laser with =248 nm and E=100 mJ) as described elsewhere.[58] The growth dynamics was investigated by monitoring the RHEED pattern. During the growth, the temperature was maintained at 750oC. Without particularly mentioned, all LMO films were grown in an oxygen environment 13 at a pressure of 1×10-3 mbar. After the deposition, the samples were slowly cooled", + "mJ) as described elsewhere.[58] The growth dynamics was investigated by monitoring the RHEED pattern. During the growth, the temperature was maintained at 750oC. Without particularly mentioned, all LMO films were grown in an oxygen environment 13 at a pressure of 1×10-3 mbar. After the deposition, the samples were slowly cooled down to room temperature at the growth pressure without further post-annealing. STEM-EELS Measurements: Microstructures of cross-sectional samples were investigated using an aberration-corrected STEM Titan at 300 keV. Elemental analysis was carried out using STEM-EELS spectrum imaging with a Gatan Quantum 966 system. The near-edge fine structures were used to study the local electronic structures at a sub-uc level of resolution with an energy resolution of 0.1 eV. Magnetic Measurements: A Quantum Design SQUID measurement system was used to measure the magnetic properties from 10 to 300 K with the magnetic field applied in-plane along the (100) direction of the substrate. The temperature-dependent magnetic measurements were performed by first cooling in 1 T, and the in-plane magnetic moments were then measured during warm-up in 0.1 T.[18] X-Ray Absorption Measurements: XAS and XMCD at the Ti and Mn L2,3 absorption edges were performed on Beamline I06 at the Diamond Light Source, UK. Oppositely circular polarized X-rays with 100% polarization degree were used successively to resolve XMCD signals from Mn and Ti. The light-helicity was switched in a saturating magnetic field of 1 T, which was applied at 60o with respect to the film plane and in parallel with the incident beam. XAS was obtained in total electron yield (TEY) mode at 10 K. The probing thickness was ~3- 5 nm. XMCD was calculated as + - - / ( )        , where  represents the TEY-XAS intensity for the respective helicities of the emitted light. Supporting Information Supporting Information is available from the Wiley Online Library or from the author. Acknowledgements W. Niu, W. Q. Liu, M. Gu, and Y. D. Chen contributed equally to this work. X. F. Wang and Y. Z. Chen conceived the study and designed the experiments. W. Niu and Y. D. Chen prepared and characterized the samples. M. Gu and P. Wang performed the STEM/EELS 14 measurements. W. Q. Liu, X. Q. Zhang, and Y. B. Xu conducted the synchrotron measurements. M. H. Zhang and Y. Q. Chen performed the partial control experiments. J. Wang and J. Du carried out the SQUID measurements. F. Q. Song, X. Q. Pan, N. Pryds, and R. Zhang contributed to the data analysis. W. Niu, X. F. Wang and Y. Z. Chen wrote the paper. This work was financially supported by the National Key Basic Research Program of China under Grant Nos. 2014CB921103, 2015CB654901 and 2017YFA0206304, the National Natural Science Foundation of China under Grant Nos. 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B 2013, 88, 054111. [58] W. Niu, M. Gao, X. Wang, F. Song, J. Du, X. Wang, Y. Xu, R. Zhang, Sci. Rep. 2016, 6, 26081. 17 LSAT LMO/LSAT LMO/LAO LAO STO LMO/STO (c) (e) (f) (d) 0 200 400 600 800 LMO/STO LMO/LSAT Intensity (a.u.) Time (s) LMO/LAO LaMnO3 Substrates (a) (b) Figure 1. Layer-by-layer epitaxial growth of LMO films on STO, LSAT and LAO substrates. (a) Sketch of the LMO-based perovskite heterostructures. (b) Representative RHEED intensity oscillations for the typical 20-uc LMO film on various substrates. (c-e) RHEED patterns of STO, LAO and LSAT prior to the growth and 20-uc LMO films after the growth on these substrates. (f) Surface morphology (2 m × 2 m) of LMO films with thickness of 20 uc on STO. The scale bar is 500 nm. 18 1×10 -7 mbar 1×10 -6 mbar 1×10 -4 mbar 1×10 -3 mbar 0 100 200 300 0 4 8 12 16 20 PO2 (mbar) TC (K) M (B/Mn) tLMO=10uc 10 -7 10 -6 10 -5 10 -4 10 -3 0.0 0.5 1.0 1.5 75 150 0 100 200 300 0 10 20 30 40 a b c 20 uc 15 uc 10 uc 7 uc 6 uc 5 uc 3 uc H=1000 Oe 5 10 15 20 0.0 0.4 0.8 1.2 1.6 0 100 200 300 0 10 20 30 40 -10 -5 0 5 10 -2 -1 0 1 2 LMO/LAO LMO/LSAT", + "40 a b c 20 uc 15 uc 10 uc 7 uc 6 uc 5 uc 3 uc H=1000 Oe 5 10 15 20 0.0 0.4 0.8 1.2 1.6 0 100 200 300 0 10 20 30 40 -10 -5 0 5 10 -2 -1 0 1 2 LMO/LAO LMO/LSAT 0H (kOe) M (B/Mn) LMO/STO tLMO=20 uc LMO/LSAT LMO/LAO LMO/STO T=10K M (nano A m2) T (K) t (uc) Ms (B/Mn) (b) (c) M (nano A m2) T (K) M (nano A m2) T (K) (a) (d) LMO/STO LMO/STO Figure 2. Emergent ferromagnetism in LMO-based heterostructures. (a) Temperature dependence of magnetic moments of LMO/STO heterostructures with the thickness ranging from 3 to 20 uc. The inset shows the magnetization measurement configuration. (b) Thickness-dependent saturation magnetic moment of LMO/STO heterostructure at 10 K. (c) Magnetic moment of 20-uc LMO films as a function of temperature grown on STO, LSAT and LAO substrates, respectively. The inset shows the M-H curves of LMO grown on these substrates. (d) Magnetic moment as a function of temperature of 10-uc LMO films grown under different PO2. The inset shows the Ms and TC as a function of the PO2. 19 630 635 640 645 650 655 660 1×10 -3 mbar 1×10 -7 mbar Energy Loss (eV) Intensity (a.u.) (g) Mn-L2,3 edge 530 540 550 1×10 -3 mbar 1×10 -7 mbar Intensity (a.u.) Energy Loss (eV) (c) O-K edge 635 640 645 650 655 660 L2 L3 Interface surface Energy Loss (eV) Intensity (a.u.) (f) Mn-L2,3 edge Mn2+ Mn3+ Mn4+ 640 650 660 (d) 640 650 660 Mn-L2,3 edge 1×10-3 mbar Mn-L2,3 edge (f) [001] [010] (a) (b) 110-3 mbar 110-7 mbar 1×10-7 mbar Mn-L2,3 edge (e) Figure 3. Valence state variations of Mn ions at different PO2. (a) and (b) Cross-sectional STEM-HAADF images of 10 uc-LMO/STO grown at 1×10-3 mbar and 1×10-7 mbar, respectively. The scale bar is 1 nm. (c) The EELS profile comparison of O-K edge of 10-uc LMO grown at the different oxygen pressure. (d) and (e) EELS mapping of Mn-L2,3 edge of each layer in the selected interfacial area indicated in (a) and (b). (f) Corresponding EELS profiles of Mn-L2,3 edge of (d). (g) Comparison of Mn-L2,3 edge EELS spectra between 10-uc LMO films grown at the different oxygen pressure. 20 455 460 465 470 L2 6 uc 5 uc 3 uc STO L3 Ti 640 645 650 655 660 L2 Mn 4+ Mn 3+ 3 uc 5 uc 6 uc 7 uc 8 uc Mn 2+ L3 Mn (a) TEY signal (c) Photon Energy (eV) Intensity (a.u.) Photon Energy (eV) Intensity (a.u.) (b) Figure 4. Mn and Ti XAS spectra of LMO/STO heterostructures. (a) The schematic diagram of the experimental configuration for the XAS/XMCD measurements. (b) Normalized isotropic XAS spectra at Mn-L2,3 edges of LMO/STO heterostructures at 10 K and 1 T. The marked four dashed lines from left to right indicate the peak positions of Mn2+, Mn4+, Mn3+ and Mn4+, respectively. (c) Ti-L2,3 edge XAS spectra of 3-uc, 5-uc and 6-uc LMO on STO. 21 5 10 15 20 0.0 0.4 0.8", + "of LMO/STO heterostructures at 10 K and 1 T. The marked four dashed lines from left to right indicate the peak positions of Mn2+, Mn4+, Mn3+ and Mn4+, respectively. (c) Ti-L2,3 edge XAS spectra of 3-uc, 5-uc and 6-uc LMO on STO. 21 5 10 15 20 0.0 0.4 0.8 1.2 1.6 XMCD SQUID 0.0 0.4 0.8 1.2 1.6 635 640 645 650 655 660 -2 0 2 4 6 integrated XMCD×0.4 Int. XAS Int. XMCD XMCD + - integrated XAS×0.2 15 uc LMO 10 K Mn-L2,3 Photon Energy (eV) Thickness (uc) Intensity (a.u.) M_SQUID (B/Mn) (a) (b) M_XMCD (B/Mn) Figure 5. Magnetic contribution of Mn ions. (a) Typical XAS/XMCD and their integration spectra of 15 uc-LMO at Mn-L2,3 edges at 10 K. Sum rules were used to calculate the average moment of Mn. (b) Thickness-dependent XMCD-derived and SQUID-derived magnetic moments at 10 K. 22 0 5 10 15 20 25 0 1 2 3 4 Ref. 18 LMO Ref. 52 Ref. 53 Ref. 54 tc≈5 uc La0.7Sr0.3MnO3 LaMnO3 STO Dead layer LMO (a) (b) (c) eg 1 eg 2 O 2p Thickness (uc) Magnetization (B/Mn) e STO LMO La0.67Sr0.33MnO3 O 2p Figure 6. Electronic reconstruction and dead-layer behavior in manganite films. (a) The schematic of the band diagram of electronic reconstruction at the interface of LMO/STO. (b) Sketch of the dead layer at the interface of LMO/STO. (c) The thickness-dependent saturated magnetization for LMO (Ref. [18] and our data) and LSMO films (Refs. [52-54]). A critical thickness of ~2 nm (5 uc) for the suppression of magnetization in the vicinity of the interface is often observed. 23 Supporting Information -10 -5 0 5 10 -2 -1 0 1 2 M (B/Mn) 0H (kOe) 10K 20K 50K 100K 200K 300K LMO/STO tLMO=20 uc Figure S1. Bulk magnetic properties of the LMO films. M(H) hysteresis loops for 20-uc LMO grown on STO at 1×10-3 mbar oxygen pressure at various temperatures. 0 1 2 3 4 5 6 7 0.0 0.5 1.0 La Mn Ti DF Mn Ti La Mix Intensity (a.u.) Position (nm) a b Figure S2. Intermixing in LMO/STO heterostructures. (a) Elemental profiles obtained from the EELS maps of the 10-uc LMO. (b) EELS elemental map with Mn in red, Ti in blue and La in yellow and the mixed image. At the interface, Mn/Ti signal can be found in a few atomic layers of LMO/STO, while La appears in the deeper regions inside the STO layer. The LMO layer is chemically wider in the La image than those in the Mn and Ti maps. An intermixing of Mn and Ti signals can be also found within a few atomic layers of LMO/STO, while La appears in the slightly deeper regions inside the STO layers. This asymmetry cation intermixing is widely observed at the interface regions between manganites and STO.[1-3] 24 455 460 465 470 L2 eg L2 t2g L3 t2g L3 eg Ti L-edge Intensity (a.u.) (b) Energy Loss (eV) T1 T8 M1 LMO M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 T1 T2 T3 T4 T5 T6 T7", + "at the interface regions between manganites and STO.[1-3] 24 455 460 465 470 L2 eg L2 t2g L3 t2g L3 eg Ti L-edge Intensity (a.u.) (b) Energy Loss (eV) T1 T8 M1 LMO M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 T1 T2 T3 T4 T5 T6 T7 T8 STO HAADF 635 640 645 650 655 660 L2 L3 M1 M11 Energy Loss (eV) Intensity (a.u.) (c) Mn L-edge Mn2+ Mn3+ Mn4+ 1 3 5 7 9 11 0 20 40 60 80 100 Mn 2+ Mn 3+/Mn 4+ Position (uc) Ratio (%) (d) (a) Figure S3. EELS analysis of Mn valence states of the LMO film grown at PO2=1×10-3 mbar. (a) HAADF image of the corresponding area indicated by the red box in Figure 3a. The green dashed line indicates the interface between LMO and STO. (b) EELS profile of Ti-L2,3 edges of each layer from T1 to T8. (c) EELS profile of Mn-L2,3 edges of each layer in the selected interfacial area. (d) Different ratios of Mn2+ and Mn3+/Mn4+ fitted from the EELS profiles in each layer. To determine the Mn3+ and Mn4+ ionic content in each sample, we simultaneously and self-consistently fit our experimental data using spectra calculated for the respective Mn species via multiplet simulations. It shows that a significant fraction of Mn2+ dominates at the interface and then remains a constant ratio. Otherwise, Mn3+/Mn4+ increases firstly and then remains unchanged with increasing the thickness of LMO. All EELS curves of Ti-L2,3 edges show a similar sharp multiplet structure. Remarkably, there is no peak shift in the detection limit of EELS, suggesting that there is no variation of Ti valence state. Contrary to a previous analysis of Ti edge yielding a change in the valence of Ti3+ below the Ti4+ state,[4, 5] the valence state of each-uc Ti in LMO/STO keeps the nominal Ti4+ of the bulk STO even in the intermixing region of LMO/STO. The unchanged valence state of Ti4+ in our LMO/STO heterostructures is also consistent with the previous STEM- EELS measurements of La1-xSrxMnO3/STO (0 ≤ x ≤ 0.5) heterostructures.[3] 25 0 1 2 3 4 5 0.0 0.5 1.0 Ti Mn La Intensity (a.u.) Position (nm) b 0.2 nm 0.2 nm [010] [001] a DF Mn Mix La Ti LMO STO c Figure S4. TEM characterization of 5-uc LMO/STO heterostructure. (a) Cross-sectional STEM-HAADF image. (b) Elemental profiles obtained from the EELS maps. At the interface, cation intermixing is also observed. (c) Dark-field (DF) image of the corresponding area indicated by the red box in panel a, and EELS elemental maps of Mn, Ti, La and the mixed image. 635 640 645 650 655 660 3 uc 5 uc 6 uc 7 uc 8 uc 15 uc Energy (eV) Intensity (a.u.) Figure S5. Mn XMCD spectra of the LMO/STO heterostructures with the varied LMO overlayer thickness. All LMO thin films are deposited at PO2= 1×10-3 mbar and the spectra are obtained at T = 10 K. 26 220 240 260 280 300 0.0 2.5x10 7 5.0x10 7 7.5x10 7 1.0x10 8 100uc 20uc 10uc", + "XMCD spectra of the LMO/STO heterostructures with the varied LMO overlayer thickness. All LMO thin films are deposited at PO2= 1×10-3 mbar and the spectra are obtained at T = 10 K. 26 220 240 260 280 300 0.0 2.5x10 7 5.0x10 7 7.5x10 7 1.0x10 8 100uc 20uc 10uc 5uc Mesurement Limit 3uc T (K) Rs (/□) Figure S6. Transport properties of LMO/STO heterostructures. Sheet resistance of 3, 5, 10, 20 and 100 uc LMO on STO as a function of temperature. All LMO films exhibit insulating behaviors and beyond the measurement limit as temperature decreases. We note that, contrary to the LAO/STO system where the interface becomes metallic above the critical thickness, our film remain insulating. This is due to the difference of band gap between LAO and LMO. The large band gap of LAO leads to the electron transfer to STO.[6] Transport properties are determined in a Van der Pauw four-probe configuration with a Quantum Design physical properties measurement system (PPMS) in the range of 2-300 K. Sum-rules The magnitude of magnetization for various-thickness LMO films is estimated quantitatively by XMCD spin sum-rule, which yields the average magnetization by using the following equations:[7] 3 2 3 2 4 ( ) 3 ( )d L L orb h L L d m n                  , (1) 3 3 2 3 2 1 6 ( ) 4 ( ) 7 (1 ) 2 ( ) L L L z spin h z L L d d T m n S d                                 . (2) where + and - are the absorption intensity with left and right circular polarized X-rays, nh is the number of holes in d shells, and in this system nh is 6.[5] is the expected value of the magnetic dipole operator and 2 is the value of mspin in Hartree atomic units, which could be omitted during the calculation.[8] Dead-layer effect in manganite thin films The dead-layer behavior is widely observed in other ferromagnetic manganite films, such as La0.7Sr0.3MnO3 (LSMO) and LMO.[9-11] The mechanism underlying this dead-layer effect remains elusive but might be due to the accumulation of Mn2+ and/or intermixing of cations or the other external effects, which always shows a critical thickness (tc) of ~2 nm (i.e., ~5 uc) for STO as the substrate. Furthermore, LAO and LSAT are both polar oxides. Growing LMO on LAO and LSAT substrates still exhibits the clear ferromagnetism although there are no polar discontinuity. This is contradictory with the previous results:[6] 12-uc LMO grown LAO measured by 27 scanning SQUID shows non-ferromagnetism. One possible reason without ferromagnetism can be attributed to the fact that 12 uc film is not thick enough beyond the dead-layer region of manganite grown on LAO and LSAT substrates. Notably, the dead-layer thickness is substrate-dependent. For example, the", + "LMO grown LAO measured by 27 scanning SQUID shows non-ferromagnetism. One possible reason without ferromagnetism can be attributed to the fact that 12 uc film is not thick enough beyond the dead-layer region of manganite grown on LAO and LSAT substrates. Notably, the dead-layer thickness is substrate-dependent. For example, the tc of the dead-layer behavior in LSMO films grown on STO, LAO and LSAT is ~8 uc, ~20 uc and ~17 uc, respectively.[10, 11] Besides, the oxygen ions in STO can diffuse over several micrometers at high temperatures during the growth.[12] This supplies more oxygen diffusing into the LMO layers, leading to the higher Ms and coercivity of LMO/STO heterostructures in comparison to the case of LMO grown on LAO and LSAT substrates. References [1] Z. Yuan, J. Ruan, L. Xie, X. Pan, D. Wu, P. Wang, Appl. Phys. Lett. 2017, 110, 171602. [2] L. F. Kourkoutis, J. H. Song, H. Y. Hwang, D. A. Muller, Proc. Natl. Acad. Sci. 2010, 107, 11682. [3] J. A. Mundy, Y. Hikita, T. Hidaka, T. Yajima, T. Higuchi, H. Y. Hwang, D. A. Muller, L. F. Kourkoutis, Nat. Commun. 2014, 5, 3464. [4] J. Garcia-Barriocanal, F. Y. Bruno, A. Rivera-Calzada, Z. Sefrioui, N. M. Nemes, M. Garcia-Hernandez, J. Rubio-Zuazo, G. R. Castro, M. Varela, S. J. Pennycook, C. Leon, J. Santamaria, Adv Mater 2010, 22, 627. [5] J. Garcia-Barriocanal, J. C. Cezar, F. Y. Bruno, P. Thakur, N. B. Brookes, C. Utfeld, A. Rivera-Calzada, S. R. Giblin, J. W. Taylor, J. A. Duffy, S. B. Dugdale, T. Nakamura, K. Kodama, C. Leon, S. Okamoto, J. Santamaria, Nat. Commun. 2010, 1, 82. [6] X. R. Wang, C. J. Li, W. M. Lü, T. R. Paudel, D. P. Leusink, M. Hoek, N. Poccia, A. Vailionis, T. Venkatesan, J. M. D. Coey, E. Y. Tsymbal, Ariando, H. Hilgenkamp, Science 2015, 349, 716. [7] D. Yi, J. Liu, S. Okamoto, S. Jagannatha, Y. C. Chen, P. Yu, Y. H. Chu, E. Arenholz, R. Ramesh, Phys Rev Lett 2013, 111, 127601. [8] P. Yu, J. S. Lee, S. Okamoto, M. D. Rossell, M. Huijben, C. H. Yang, Q. He, J. X. Zhang, S. Y. Yang, M. J. Lee, Q. M. Ramasse, R. Erni, Y. H. Chu, D. A. Arena, C. C. Kao, L. W. Martin, R. Ramesh, Phys. Rev. Lett. 2010, 105, 027201. [9] R. P. Borges, W. Guichard, J. G. Lunney, J. M. D. Coey, F. Ott, J. Appl. Phys. 2001, 89, 3868. [10] M. Huijben, L. W. Martin, Y. H. Chu, M. B. Holcomb, P. Yu, G. Rijnders, D. H. A. Blank, R. Ramesh, Phys. Rev. B 2008, 78, 094413. [11] R. Peng, H. C. Xu, M. Xia, J. F. Zhao, X. Xie, D. F. Xu, B. P. Xie, D. L. Feng, Appl. Phys. Lett. 2014, 104, 081606. [12] W. Niu, Y. Gan, Y. Zhang, D. V. Christensen, M. v. Soosten, X. Wang, Y. Xu, R. Zhang, N. Pryds, Y. Chen, Appl. Phys. Lett. 2017, 111, 021602.", + "[12] W. Niu, Y. Gan, Y. Zhang, D. V. Christensen, M. v. Soosten, X. Wang, Y. Xu, R. Zhang, N. Pryds, Y. Chen, Appl. Phys. Lett. 2017, 111, 021602.", + "1 Room temperature observation of electron resonant tunneling through InAs/AlAs quantum dots Jie Suna), Ruoyuan Li, Chang Zhao, Like Yu, Xiaoling Ye, Bo Xu, Yonghai Chen and Zhanguo Wangb) Key Laboratory of Semiconductor Materials Science, Institute of Semiconductors, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, China Molecular beam epitaxy is employed to manufacture self-assembled InAs/AlAs quantum-dot resonant tunneling diodes. Resonant tunneling current is superimposed on the thermal current, and they make up the total electron transport in devices. Steps in current-voltage characteristics and peaks in capacitance-voltage characteristics are explained as electron resonant tunneling via quantum dots at 77K or 300K, and this is the first time that resonant tunneling is observed at room temperature in III-V quantum-dot materials. Hysteresis loops in the curves are attributed to hot electron injection/emission process of quantum dots, which indicates the concomitant charging/discharging effect. Electrochemical and Solid State Letters 9 G167-170 2006 a) Electronic mail: albertjefferson@sohu.com Present at: Solid State Physics, Lund University, P. O. Box 118, SE-221 00 Lund, Sweden b) Electronic mail: zgwang@red.semi.ac.cn 2 Self-assembled quantum dots (QDs) grown in Stranski-Krastanow (S-K) mode have attracted much interest due to their potential applications in novel nanoscale devices1. However, with their optical properties having been well clarified, relatively less researches are done on transport through QDs, such as the fabrication of QD resonant tunneling diodes (RTDs). RTDs have the advantage over conventional circuits2 in terms of reduced circuit complexity for implementing a given function. Since the quantum dots confine electrons in all three dimensions, resonant tunneling into the quantized states will give rise to additional peaks in current-voltage (I-V) characteristics3 compared with quantum well RTDs. Furthermore, QD RTDs will operate with low power at high speed on account of their small sizes of devices. Therefore, QD RTDs should have a more promising future than quantum well RTDs. To date, however, all commercial RTDs are made of quantum wells whilst much more efforts are needed to make the quantum-dot RTDs practical. Room temperature observations of electron resonant tunneling via Si/SiO2 QDs4 or InGaN/GaN QDs5 have been successfully achieved by virtue of the very large barrier height of matrix materials. However, a clear current bump due to resonant tunneling through III-V QDs e.g. InAs QDs can only be detected at extremely low temperatures6-9, typically 1.6K-130K. Besides, nearly all this kind of experiment is focused on current-voltage characteristics of the samples, whereas little attention is paid to capacitance-voltage (C-V) characteristics. In the present letter, for the first time, clear step structures 3 in the I-V curves of QD RTDs are observed at room temperature, and explained in terms of resonant tunneling via an ensemble of InAs/AlAs QDs. Also studied by C-V measurements here is the concomitant resonant charging effect. Hysteresis loops in I-V and C-V curves provide clues of hot electron charging and discharging procedure, thus further convincing us that as-grown samples show quantum-dot behavior. This finding represents a great step forward towards the applications of QD RTDs. Our RTD samples, shown schematically in Fig. 1 (a), are grown by a RIBER 32P solid-source molecular beam epitaxy", + "clues of hot electron charging and discharging procedure, thus further convincing us that as-grown samples show quantum-dot behavior. This finding represents a great step forward towards the applications of QD RTDs. Our RTD samples, shown schematically in Fig. 1 (a), are grown by a RIBER 32P solid-source molecular beam epitaxy (MBE) machine on GaAs (100) substrate (n+=2×1018cm-3). The growth is started at substrate temperature Ts of 600oC by depositing a 300nm n-GaAs, in which the Si concentration is graded from 2×1018cm-3 to 1×1017cm-3, and a 10nm undoped GaAs spacer layer. Then, a 5nm undoped AlAs barrier is grown and Ts lowered to 530oC. At that point, nominally 1.9 monolayers (ML) InAs are deposited with a very slow rate of 0.015ML/s (the In delivery is cycled in 1.7s of evaporation followed by 5.0s of interruption until the given InAs thickness is reached) and a 2 min post-growth annealing10 is processed. Low deposition rate, growth interruption and post-growth annealing all help to equilibrate the surface by enhancing the migration of In adatoms11, consequently producing high quality QDs. After the QDs are capped by another 5nm undoped AlAs, Ts is increased to 600oC. 10nm undoped GaAs spacer and 300nm graded n-GaAs (1×1017cm-3 to 4 2×1018cm-3) are deposited in turn. With standard photolithography and wet etching procedures being used to obtain 60μm×60μm mesas, alloyed Au/Ge/Ni contacts are fabricated via small windows (40μm×40μm) in 350nm SiO2 insulation layers. The sheet density of QDs is ~6×1010cm-2 as determined from plan-view transmission electron microscope (TEM) measurements. It is known that upon AlAs capping at high growth temperature, In atoms will segregate towards the exterior to lower surface energy (the segregation ratio is 0.77 at 530oC, which is only slightly smaller than 0.85 in the InAs/GaAs system12). The upper AlAs barrier in our device, therefore, is in fact InxAl1-xAs ternary alloy. It is for this reason that the height of upper barrier has been lowered to some extent, as is illustrated schematically in Fig. 1 (b). Such effects will greatly influence the electronic properties of RTDs, which will be set forth thereinafter. Summarized in Fig. 2-4 are room temperature and 77K I-V and C-V characteristics of three typical samples (A, B, C). It is with a crystal-controlled 1MHz 15mV test signal that the C-V properties are measured. C-V curves at 300K are shifted by 10pF for clarity. All curves are reproducible and the arrows indicate the voltage-scanning directions. All of the voltage sweeping begin and end at large negative bias. For all devices, the current at forward (positive) bias is smaller than at reverse (negative) bias. These asymmetric I-V characteristics reflect the intrinsic properties of the devices, which are attributed to inhomogeneity between Ohmic contacts of both sides and 5 between the two asymmetric AlAs barriers [see Fig. 1 (b)]. Similar to previous studies8, 9, the basic shapes of our I-V curves are determined by the hot current, and the resonant tunneling current which overlies on main structures only contributes to a small fraction of total current. Resonant tunneling signals (steps in I-V and peaks in C-V), at 77K or", + "Similar to previous studies8, 9, the basic shapes of our I-V curves are determined by the hot current, and the resonant tunneling current which overlies on main structures only contributes to a small fraction of total current. Resonant tunneling signals (steps in I-V and peaks in C-V), at 77K or room temperature, are clearly observed in the figures and summarized in Table 1. Structures at 0.5―0.7V, -0.7―-1.1V are attributed to resonant tunneling through ground states of QDs, and signals at 2.1―2.8V, -1.9―-2.6V to first excited states. The cause for the broadening of I-V steps and C-V peaks is to be detected in the size undulation and lateral coupling of QDs13. It is well known that in resonant tunneling experiments, electrons will accumulate between double barriers14, and that is why one can observe peaks in C-V properties when resonance takes place. This resonant tunneling charging effect will be enhanced in asymmetric double-barrier structures, whose reasons go as follows. In asymmetric double-barrier structures15, 16, such as in our case [see Fig. 1 (b)], under forward bias the emitter barrier is less transparent than the collector one. That is, TE«TC, where TE and TC are electron transmission coefficients of emitter and collector barriers respectively. Therefore, there are relatively less electrons which remain in QDs while the majority passes through double barriers during resonant tunneling. When it comes to the reverse bias (in this case the emitter barrier is the upper InxAl1-xAs barrier), TE»TC, the electron density in the dots is considerably enlarged and resonant tunneling current drastically diminished. In 6 other words, resonant tunneling signals in I-V and C-V will form a complementary pair. In Table 1, generally speaking, under negative bias the steps in I-V are non-salient whereas the corresponding C-V peaks strong enough to survive at 300K, indicating a strong resonant tunneling charging effect. When it comes to the positive bias, on the contrary, with the I-V stairs being obvious the C-V structures are hard to be found except at ~2.2V in sample B. It means that the resonance current is increased and the resonant tunneling charging effect decreases. Compared with sample A which does not exhibit any resonant tunneling signals at small positive bias (<2V) in I-V curve, one may notice how big the resonant tunneling current in samples B, C is. The step at 0.5V in I-V curve of sample C even persists up to room temperature. As is mentioned above, the total current in our devices consists of two components: thermal current which determines the basic shape of I-V curve, and resonant tunneling current which generates steps superimposed on the main body. The latter having been discussed in detail, we will now focus on the former. I-V hysteresis loops, which are hints of hot electron charging effect (please note this is a completely different mechanism from the resonant tunneling charging effect investigated in the last paragraph) in QDs, are detected at liquid nitrogen temperature. Usually, when a sample with embedded QDs is biased, there may be two opposite consequences. One is charging of QDs via trapping hot electrons17 (viz hot", + "is a completely different mechanism from the resonant tunneling charging effect investigated in the last paragraph) in QDs, are detected at liquid nitrogen temperature. Usually, when a sample with embedded QDs is biased, there may be two opposite consequences. One is charging of QDs via trapping hot electrons17 (viz hot electrons relax into the QDs), and the other discharging 7 of QDs by direct tunneling out through the barrier18. The dominant one will be selected through the competition between the two procedures. When I-V measurements start from big reverse bias (absolute value >1V), many QDs will capture relaxed hot electrons because thermal current is quite large. The electrons can not tunnel out easily since the collector barrier is high. These negatively charged QDs induce repulsive electrostatic potentials and act as scattering centers which conspicuously reduce the following current. The current will therefore be at its low states. In this stage, the C-V peaks also appear lower, for the resonant tunneling current lays aside less electrons into QDs due to the Coulomb repulsive effect from the hot electrons captured by QDs. When large positive voltages (>2V) are reached, the captured electrons are obliged to tunnel out19 via the relatively lower collector barrier, and thereafter the current and capacitance alter to their high states in reverse scanning direction. The charging process at big negative voltage and discharging process at big positive voltage are schematically illustrated in Fig. 1 (b) by arrows (the energy band tilt under bias is ignored for simplification). The I-V hysteresis phenomena vanish at room temperature because of the limitedness of electron storage time. The electrons can easily and swiftly escape from QDs because of their large thermal energy at high temperature. Nevertheless, the loops do remain in C-V at 300K, which implies in that case the Coulomb repulsive forces come from hot electrons trapped then and there into the QDs. That conjecture can be further confirmed by the 300K C-V 8 curve of sample B. When it scans from 0V to 3.3V, since there is almost no Coulomb repulsive forces (the large-negative-bias-trapped hot electrons have already escaped due to their transient lifetime in QDs at 300K), C-V resonant tunneling charging peak at 2.2V will be relatively high. While scanning from 3.3V back to 0V, the QDs may seize electrons from the much enhanced forward hot current at 300K. Therefore, the C-V peak at 2.2V will be lowered. In contrast to the 77K status where an anticlockwise hysteresis loop α is observed, the C-V loop β at 300K is clockwise. Therefore, it can be concluded that at room temperature the repellence does not arise from large-negative-bias-trapped17 hot electrons, but from the real time trapped ones. Finally, the discussion will go to several controversial issues. a) The detected signals are not because of resonant tunneling via the X energy band valleys in AlAs barriers. In similar researches, resonant tunneling through the X states of AlAs layers is usually not observed8, 20, 21, indicating the X valleys of AlAs usually do not play a significant role in such kind of experiments. The resonance", + "of resonant tunneling via the X energy band valleys in AlAs barriers. In similar researches, resonant tunneling through the X states of AlAs layers is usually not observed8, 20, 21, indicating the X valleys of AlAs usually do not play a significant role in such kind of experiments. The resonance peaks in C-V characteristics and hysteresis loops in I-V and C-V curves all indicate a pronounced charging and discharging effect in our devices, and they strongly suggest that it is a quantum-dot behavior. b) More often than not, some abnormal phenomena, such as a swinging drop of capacitance before overflow from range, will be observed in C-V measurements when the conduction current is too large22. The mechanism is still arguable23. While the current passing 9 through devices is too large, therefore, C-V measurements seem to be unreliable. That might explain why the hot electron injection/emission effect and the resonant tunneling peaks through QDs’ first excited states (except at 2.2V in sample B) are not embodied in our C-V curves. c) In some previous researches9, 24, not all the devices show resonant tunneling behaviors. Under our conditions, resonance signals can be found in all RTDs. However, there is still a certain kind of inhomogeneity among devices. The differences among sample A, B, C are not understood at present and more tests are needed to delve into the mystery. We have, in short, fabricated InAs/AlAs QD RTDs for I-V and C-V studies. The total current passing through the devices is composed of hot electron current and resonance current. The observed stairs and peaks are attributed to electron resonant tunneling through QDs at 77K or 300K. To the best of our knowledge, this is the first direct observation of such phenomena at room temperature. Hysteresis loops in I-V or C-V curves are related to hot electron injection or emission via QDs, thus further convincing us that all the structures found in this experiment are quantum-dot behaviors. It is believed that this work will contribute to the potential applications of InAs QD RTDs. Thanks go to Professors Hongqi Xu, Jianbai Xia, Yangyuan Wang and Jilin Gao for their generous support. Part of the work is finished with the help of Mrs. Chunli Yan. The financial aid is 10 acknowledged highly valuable from Special Funds for Major State Basic Research Project of China (No 2000068303). 11 References 1 P. M. Petroff, A. Lorke, and A. Imamoglu, Phys. Today 54, 46 (2001); D. Bimberg, M. Grundmann, F. Heinrichsdorff, N. N. Ledentsov, V. M. Ustinov, A. E. Zhukov, A. R. Kovsh, M. V. Maximov, Y. M. Shernyakov, B. V. Volovik, A. F. Tsatsul’nikov, P. S. Kop’ev, and Zh. I. Alferov, Thin Solid Films 367, 235 (2000); Z. G. Wang, Y. H. Chen, F. Q. Liu, and B. Xu, J. Cryst. 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Lett. 70, 105 (1997). 22 C. H. Champness and W. R. Clark, Appl. Phys. Lett. 56, 1104 (1990). 23 M. Ershov, H. C. Liu, L. Li, M. Buchanan, Z. R. Wasilewski, and A. K. Jonscher, IEEE Trans. Electron Devices 45, 2196 (1998); J. G. Ma, K. S. Yeo, and M. A. Do, IEEE Trans. Electron Devices 46, 2357 (1999). 24 H. W. Li and T. H. Wang, Physica B 304, 107 (2001). 14 Figure and table captions Fig.1(a). Schematic picture of the epitaxially grown RTD structure. Fig.1(b). Depiction of band structure of the diode. The upper AlAs barrier is unintentionally changed into InxAl1-xAs alloy layer due to the In segregation effect during MBE growth. This makes the double barriers asymmetric. The upper arrow indicates the hot electron injection process at large negative voltage. The lower arrow indicates the hot electron tunneling out (i.e. emission) procedure at large positive voltage. Note the energy band tilt under bias is ignored in this figure for clarity. Fig.2. Current-voltage and capacitance-voltage characteristics of sample A at 77K or 300K. The directions of voltage sweeping are shown by arrows. The scan begins and ends at -3V. Fig.3. I-V and C-V curves of sample B. Hysteresis in the curves are caused by hot electron charging/discharging process. Note that loop α and β are of opposite directions. Fig.4. I-V and C-V curves of sample C. The I-V curves are dominated by thermal current, with resonant tunneling current being superimposed on. The C-V peaks are due to so called resonant tunneling charging effect. Table 1. Resonant tunneling signals in I-V and C-V properties of three typical samples at different biases. G and E stand for ground states and first excited states of QDs respectively. In general, I-V and C-V signals form a complementary pair. 15 Figure 1 (a) Figure 1 (b) 16 Figure 2 17 Figure 3 18 Figure 4 19 Forward bias Reverse bias I-V C-V I-V C-V 77K 300K 77K 300K 77K 300K 77K 300K Unit: V G E G E G E G E G E G E G E G E Sample A — 2.5 — — — — — — — — — — -1 -2.6 -1.1 — Sample B 0.7 2.1 — — — 2.2 — 2.2 -0.7 — — — -0.7 -1.9 -0.7 — Sample C 0.5 2.8 0.5 — — — — — -0.8 — — — -1 -2.1 -0.9 — Table 1", + "physica status solidi Electronic quantum optics beyond the integer quantum Hall effect Dario Ferraro*,1, Thibaut Jonckheere1, J´erˆome Rech1, Thierry Martin1 1 Aix Marseille Univ, Univ Toulon, CNRS, CPT, Marseille, France Received XXXX, revised XXXX, accepted XXXX Published online XXXX Key words: Nanophysics, quantum noise, single electron sources, integer/spin quantum Hall effect, Andreev reflection. ∗Corresponding author: e-mail Dario.Ferraro@cpt.univ-mrs.fr The analog of two seminal quantum optics experiments are considered in a condensed matter setting with sin- gle electron sources injecting electronic wave packets on edge states coupled through a quantum point contact. When only one electron is injected, the measurement of noise correlations at the output of the quantum point con- tact corresponds to the Hanbury-Brown and Twiss setup. When two electrons are injected on opposite edges, the equivalent of the Hong-Ou-Mandel collision is achieved, exhibiting a dip as in the coincidence measurements of quantum optics. The Landauer-B¨uttiker scattering theory is used to first review these phenomena in the integer quantum Hall effect, next, to focus on two more exotic systems: edge states of two dimensional topological in- sulators, where new physics emerges from time reversal symmetry and three electron collisions can be achieved; and edges states of a hybrid Hall/superconducting de- vice, which allow to perform electron quantum optics experiments with Bogoliubov quasiparticles. Copyright line will be provided by the publisher 1 Introduction. Electronic quantum optics (EQO) [1, 2] aims at exploring the intimate nature of solid states sys- tems by generating, manipulating and measuring individ- ual electronic wave-packets (WPs) ballistically propagat- ing in mesoscopic devices, in the same spirit as what is conventionally done for photons transmitted along wave- guides. This opens new prespectives for real time electron interferometry in the context of nanophysics. For this pur- pose, on-demand single electrons and holes sources (SES) have been achieved, for instance by means of driven meso- scopic capacitors [3,4,5,6] coupled via a quantum point contact (QPC) to the edge states of an integer quantum Hall (IQH) system, or via properly designed Lorentzian volt- age pulses [7,8,9] imposed on a two dimensional electron gas. The edge states in the IQH effect, which are exempt of backscattering, play the role of such wave guides and a QPC placed downstream is equivalent to a half-silvered mirror, as it partitions electrons which are either reflected by or transmitted through the QPC. Electrons differ from photons in many aspects: they obey fermionic statistics, they are charged particles which interact among them- selves and with their environment and, finally, they are al- ways accompanied by a Fermi sea close to which electron hole pairs may be easily generated. Moreover, while in quantum optics experiments the coincidence rate is mea- sured at the two outputs, here the noise cross-correlation signal at zero frequency is typically computed or measured [2]. One of the main achievement of EQO has been the real- ization of the electronic equivalent of the Hanbury-Brown and Twiss [10] (HBT) experiment where a single electron source injects electrons on a QPC, followed by the Hong- Ou-Mandel [11] (HOM) experiment, where two electrons incident from two independent sources collide", + "of the main achievement of EQO has been the real- ization of the electronic equivalent of the Hanbury-Brown and Twiss [10] (HBT) experiment where a single electron source injects electrons on a QPC, followed by the Hong- Ou-Mandel [11] (HOM) experiment, where two electrons incident from two independent sources collide at the QPC. These scenarios represent fundamental tests of quantum mechanics for both photons and electrons, as they probe both their statistics and the form of the injected WPs. In the electronic HBT interferometer the Pauli principle leads to the anti-bunching between the injected electrons and the thermal ones incoming from the second channel [12]. In the HOM fermionic setup, when the emissions of the two colliding electrons are perfectly synchronized, one expects a suppression of the noise due again to the Pauli princi- ple because the electrons are forced to emerge on oppo- site sides of the QPC. Conversely, for a long enough de- lay, twice the HBT signal is obtained for the noise in the Copyright line will be provided by the publisher arXiv:1610.01043v1 [cond-mat.mes-hall] 4 Oct 2016 2 Dario Ferraro et al.: Electronic quantum optics... V1HtL V2HtL I2 outHtL I1 outHtL Figure 1 (Color online) HOM setup in the IQH regime. Two chiral edge channels meet at a QPC. Each one is cou- pled to a SES in the optimal regime of emission of elec- trons (holes). Current cross-correlations at the two outputs are measured at zero frequency as a function of the delay between the electron emissions. Picture taken from [13]. HOM setup as the two sources are independent [13]. Ex- periments [14] do validate the presence of a Pauli dip in the noise correlations, although this dip does not fall to zero at coincident injection: this is attributed to decoherence ef- fects because of the presence of two or more interacting channels [15]. In this paper, we wish to point out that EQO is not limited to the IQH regime. In addition to presenting the basic physics of HOM interferometry for IQH at filling factor ν = 1, we explore two different situations where the paradigms of EQO are also present. First, we con- sider the situation of the quantum spin Hall (QSH) effect in a two dimensional (2D) topological insulator [16,17], where two counter-propagating edge states carrying elec- trons with opposite spin appear on each side of the Hall bar [18]. In this situation (with or without spin flip processes at the QPC) it is possible to study the interplay between Fermi statistics and time reversal symmetry (TRS) using HOM interferometry, with two or even three SES. Next, we study the interplay between the IQH effect and supercon- ductivity, as Andreev reflections convert an injected elec- tron into a coherent superposition of electrons and holes – a Bogoliubov quasiparticle – which can subsequently be used as an input quasiparticle for an HBT or an HOM in- terferometric device [19]. Such Bogoliubov quasiparticles, although generated from electrons of finite energy above the Fermi sea, have recently been identified as possible re- alization of Majorana fermions in a", + "a Bogoliubov quasiparticle – which can subsequently be used as an input quasiparticle for an HBT or an HOM in- terferometric device [19]. Such Bogoliubov quasiparticles, although generated from electrons of finite energy above the Fermi sea, have recently been identified as possible re- alization of Majorana fermions in a condensed matter sys- tem [20,21,22]. For simplicity, in both of these extensions, we work solely with scattering theory and do not include interactions between the edges or with the electromagnetic environment surrounding the device. 2 Hong-Ou-Mandel electron collisions in the in- teger quantum Hall effect. We start by considering the case of standard IQHE in the QPC geometry [23] (see Fig. 1). A SES injects electrons on each incoming edge state, and current cross-correlations are detected at the two outputs of the QPC, exhibiting a dependence on the de- lay between injections. In experiments, the voltage drive is typically a periodic square wave, which results in the con- trolled emission of a regular train of single electrons and holes [3,4,24]. We consider the outgoing current cross-correlations at zero frequency: S out 12 = Z dt dt′ I out 1 (t)I out 2 (t′) c (1) where I out 1 (t) and I out 2 (t) are the currents outgoing from the QPC (see Fig. 1) and where we defined the connected correlator ⟨AB⟩c = ⟨AB⟩−⟨A⟩⟨B⟩. We can safely as- sume a linear dispersion for the electrons along the edge and chirality (from now on we assume the Fermi velocity vF = 1, reintroducing it only where needed). We com- pute the cross-correlations at the output of the QPC, using conventionally a x-axis pointing in the propagation direc- tion of each edge. A scattering matrix with transmission (reflection) probability T (R = 1 −T ) characterizes the QPC. The currents outgoing from the QPC can be written in terms of the incoming field operators as I out 1 = T I1 + RI2 + ie √ RT (Ψ † 1Ψ2 −Ψ † 2Ψ1) I out 2 = RI1 + T I2 −ie √ RT (Ψ † 1Ψ2 −Ψ † 2Ψ1), with Ψl (l = 1, 2) the annihilation operator for an electron on edge l and where we neglected the time dependence for notational convenience. Replacing these expressions into Eq. (1) allows to express the outgoing noise as [25] S out 12 = RT (S11 + S22 + Q) . (2) The last term encodes quantum interference contributions: Q = −e2 Z dtdt′ h ⟨Ψ1(t)Ψ † 1(t′)⟩⟨Ψ † 2(t)Ψ2(t′)⟩ + ⟨Ψ † 1(t)Ψ1(t′)⟩⟨Ψ2(t)Ψ † 2(t′)⟩ i , (3) while the first two terms are the incoming auto-correlation noise associated respectively to I1 and I2. Notice that the averages are evaluated with respect to the initial state |ϕ⟩ and correspond to the first order electronic coherence func- tions defined in [25]. Calculations are performed in the simple case in which a single electron, with a given WP, is injected into each edge. This allows to obtain analytical expressions for the HOM dip, which corresponds to the cross-correlated noise when two WPs collide from opposite", + "electronic coherence func- tions defined in [25]. Calculations are performed in the simple case in which a single electron, with a given WP, is injected into each edge. This allows to obtain analytical expressions for the HOM dip, which corresponds to the cross-correlated noise when two WPs collide from opposite sides of the QPC. Copyright line will be provided by the publisher pss header will be provided by the publisher 3 The state corresponding to an electron injected with WP ϕ1,2(x) on edge 1, 2 reads: |ϕ1,2⟩= Z dx ϕ1,2(x) Ψ † 1,2(x) |F⟩ (4) where |F⟩indicates the edge state Fermi sea at finite tem- perature T. For identical WPs ϕ1,2 = ϕ(x), reaching the QPC with a delay δt, the noise normalized by twice the HBT noise (only one source emitting [12]) is: SHOM(δt) 2SHBT = 1 − R ∞ 0 dk| ˜ϕ(k)|2e−ikδt(1 −fk)2 R ∞ 0 dk| ˜ϕ(k)|2(1 −fk)2 2 , (5) where fk = 1 1 + e(k−kF )/T (6) is the Fermi distribution (kF the Fermi momentum), ˜ϕ(k) is the WP in momentum space, and SHBT = −e2RT R ∞ 0 dk| ˜ϕ(k)|2(1 −fk)2 R ∞ 0 dk| ˜ϕ(k)|2(1 −fk) !2 . (7) Eq. (5) shows immediately that SHOM(0)/2SHBT = 0, as a consequence of Fermi statistics. On the opposite, when δt is much larger than the inverse width of ˜ϕ(k), SHOM(∞) is the sum of the HBT noise of the two electrons taken in- dependently, and SHOM/(2SHBT ) = 1. At low tempera- ture, when the injected WP does not overlap with the Fermi sea, one has: SHOM(δt) 2SHBT = 1 − Z dτ ϕ(τ) ϕ∗(τ + δt) 2 , (8) which is similar to the case of optics [11] as the profile of the HOM dip corresponds to the auto-convolution of the WP. The functional forms of the HOM dip can be obtained analytically for various types of WPs. The SES of Ref. [3] is believed to generate Lorentzian WPs of the form: ˜ϕ(k) = √ Γ √ 2π 1 (k −k0) + i Γ 2 (9) which corresponds to Breit-Wigner resonance associated with the emission by the discrete level of a quantum dot of width Γ/2 at energy k0, with a half exponential (see Fig 1) real space profile ϕ(x) = √ Γeik0xe Γ 2 xθ(−x) (θ(x) is the Heaviside function). At zero temperature, the noise corre- sponding to two such WPs, centered at the same energy k0 but with different widths Γ1,2 reads: SHOM(δt) 2SHBT = 1− 4Γ1Γ2 (Γ1 + Γ2)2 h θ(δt)e−Γ1δt+θ(−δt)eΓ2δti . (10) This HOM dip lacks mirror symmetry: its exponen- tial behavior is characterized by the time constants Γ −1 1 -100 -50 0 50 100 0.0 0.2 0.4 0.6 0.8 1.0 0 T 0 0 1 ∆t SHOM H∆tL 2 SHBT -100 -50 0 50 100 0.0 0.2 0.4 0.6 0.8 1.0 ∆t SHOM H∆tL 2 SHBT Figure 2 (Color online) Normalized HOM noise as a func- tion of the delay in the electron (hole) emission δt. Full curves represent the result obtained within the Floquet scattering matrix formalism in", + "SHBT -100 -50 0 50 100 0.0 0.2 0.4 0.6 0.8 1.0 ∆t SHOM H∆tL 2 SHBT Figure 2 (Color online) Normalized HOM noise as a func- tion of the delay in the electron (hole) emission δt. Full curves represent the result obtained within the Floquet scattering matrix formalism in the optimal emission regime of the SES. Dotted-dashed curves are the analytical predic- tions of Eq. (10) for exponential WPs. Upper panel: sym- metric profile, with emitter transparencies D = 0.2 (red curve), 0.5 (green curve) and 0.8 (blue curve). Inset: dips for D = 0.2 on two periods of the applied voltage. Lower panel: asymmetric profile, with transparencies D1 = 0.2, D2 = 0.5 (bottom red curve) and D1 = 0.1, D2 = 0.8 (top blue curve). Other parameters are: T0 = 400 (in units of ¯h/∆) the period of the applied voltage, and T = 0.01∆ the temperature. Picture taken from [13]. or Γ −1 2 respectively depending on the sign of δt. The dip does not reach zero as for Eq. (5). The reduced contrast (less than unity) reflects the distinguishability of the in- jected electrons. This asymmetry is clearly not present for WPs generated by a Lorentzian voltage applied directly to the edge channels [7,9,26] or when considering the adia- batic limit for the emission of a SES [23,27,28]. We compare our formulas with the numerical results of a Floquet calculation [29,30,31] properly modeling the emission process from a realistic periodic source [6,24]. This emitter consists of the mesoscopic capacitor [3,4,12]: a quantum dot with discrete levels connected through a QPC to the edge state which is driven by a gate applying a periodic square drive V (t). The highest occupied state is first positioned above the Fermi level, causing the tunnel- ing of a dot electron to the edge; this (now) empty level is next placed below the Fermi sea, causing the emission of a hole. Periodic square voltage with an amplitude identical to the dot level spacing ∆yield optimal emission. Copyright line will be provided by the publisher 4 Dario Ferraro et al.: Electronic quantum optics... The Floquet approach allows to evaluate numerically current and noise [24]. Fig. 2 shows the comparison be- tween the numerical results for the HOM dip with the ana- lytical formula in Eq. (10). The upper panel shows a sym- metric profile, due to the fact that the two SES are iden- tical. The dips (with minimum value 0 at zero delay) are broader for lower emitter transparencies. This is a conse- quence of the fact that electrons take a longer time to exit the dot leading to a broader WP. In the optimal regime, the electron emission time corresponds to [24,32]. τ = 2π ∆ \u0012 1 D −1 2 \u0013 . (11) This value (with Γ = τ −1) is chosen to plot the ana- lytical predictions from Eq. (10) (dotted-dashed curves). The agreement is excellent, especially at low enough trans- parency, where the single electron emission is properly achieved [4,25]. For what it concerns the asymmetric pro- file", + "\u0013 . (11) This value (with Γ = τ −1) is chosen to plot the ana- lytical predictions from Eq. (10) (dotted-dashed curves). The agreement is excellent, especially at low enough trans- parency, where the single electron emission is properly achieved [4,25]. For what it concerns the asymmetric pro- file (lower panel), the contrast is smaller than 1. Also in this case both the numerical and the analytical approaches agree very well. The HOM interferometry with fermions is thus char- acterized by a dip in the zero-frequency current cross- correlations in collisions between two electrons, whose shape depends on the injected WPs. This same setup also offers the interesting possibility (not shown) to achieve electron hole collisions, which produce an HOM peak at large enough temperature [13]. 3 Electronic quantum optics with 2D topological insulators. Such materials exhibit the QSH effect [16, 17]. The first experimental observations of this peculiar state of matter have been carried out in CdTe/HgTe [33, 34] and InAs/GaSb [35,36,37] quantum wells. QSH effect manifests itself through a gapped bulk and metallic edge states [38] where electrons with opposite spin propagate in opposite directions along the edges as a consequence of spin-orbit interaction. The topologically protected heli- cal edge states of QSH effect, with their spin-momentum locking properties, suggests that they could be studied in a EQO context [18] especially given the recent proposals for an electron source and beam-splitter. Indeed, the character- ization of the SES has already been provided [39,40]. This pair electron source (PES) are predicted to trigger the in- jection into the helical edge states of a pair of electrons (holes) with opposite spin per period. However, the ex- perimental realization of a QPC in the QSH regime still represents a challenge due to the same Klein mechanism which prevents confinement of massless Dirac fermions in graphene [41]. Still, new generation heterostructures [36, 37] give reasonable hopes of possible applications to EQO. 3.1 Model. The Hamiltonian of the system, in the presence of a QPC, is given by [40,42,43,44] H = H0 + Hsp + Hsf (12) Σ R, ↑ L, ↓ R, ↓ L, ↑ PES1 PES2 PES3 Figure 3 (Color on line) Schematic view of a QSH bar with spin up (full arrows) and spin down (dashed arrows) electrons. Dotted lines indicate the scattering processes, compatible with TRS, which affect the current in the R, ↑ outgoing channel (see main text). Pair-electrons sources (PES) are represented with colored circles. Picture taken from [18]. with H0 = −i¯h X α=R,L X σ=↑,↓ Z +∞ −∞ dxξα : Ψ † α,σ(x)∂xΨα,σ(x) : (13) the free Dirac Hamiltonian of the one-dimensional helical edge channels. Here Ψα,σ(x) extends the previous defini- tion for the electronic annihilation operator by taking into account the chirality (α = R, L) and spin (σ =↑, ↓) de- grees of freedom. Moreover, ξR/L = ±1 represents the chirality index and : ... : indicates the conventional nor- mal ordering. Assuming a local QPC, one obtains two ad- ditional contributions: Hsp = 2¯h X σ=↑,↓ γspΨ † R,σ(0)ΨL,σ(0) + h.c. (14) the", + "L) and spin (σ =↑, ↓) de- grees of freedom. Moreover, ξR/L = ±1 represents the chirality index and : ... : indicates the conventional nor- mal ordering. Assuming a local QPC, one obtains two ad- ditional contributions: Hsp = 2¯h X σ=↑,↓ γspΨ † R,σ(0)ΨL,σ(0) + h.c. (14) the spin-preserving and Hsf = 2¯h X α=R,L ξαγsfΨ † α,↑(0)Ψα,↓(0) + h.c. (15) the spin-flipping tunneling Hamiltonian. TRS of the total Hamiltonian H is guaranteed as long as γsp and γsf are real numbers [42,45] (γsp > γsf is gen- erally assumed). The evolution of the fermionic field oper- ators is specified by the Heisenberg equation of motion in Dirac form. This allows to specify the incoming/outgoing scattering states/operators, which are related by a 4 × 4 scattering matrix describing the QPC [42,46], whose struc- ture reflects the TRS. The scattering matrix elements are then parametrized by λpb, λff, λpf [42,46], namely the amplitude probabilities of spin-preserving backscattering, spin-flipping forward scattering and spin-preserving for- ward scattering processes (respectively orange, magenta and green dotted lines in Fig. 3) [40]. They naturally satisfy the constraint |λff|2 + λ2 pf + |λpb|2 = 1 (16) Copyright line will be provided by the publisher pss header will be provided by the publisher 5 imposed by charge conservation. 3.2 Auto-correlated noise. We want to investigate now the auto-correlated outgoing noise. We will focus for simplicity on the (R, ↑) channel, the expressions for the other possible cases can be derived proceeding in the same way. It reads S out R↑,R↑= Z dtdt′⟨I out R↑(t)I out R↑(t′)⟩ρ,c (17) where the notation indicates that the connected correlator is calculated over a density matrix of the form ρ = |ϕ⟩⟨ϕ|. In terms of the incoming signals, it can be written as S out R↑,R↑= |λff|4SR↓,R↓+ λ4 pfSR↑,R↑ + |λpd|4SL↑,L↑+ Q. (18) Notice that the interesting physics is encoded in the last term of Eq. (18) which extends what we already evaluated in the IQH case, while the others are the auto-correlations of the incoming currents, which vanish when taking the av- erage over one period and computing their zero frequency Fourier component [4,12]. Its explicit form here is Q = h (A + B + C)Q(F S) + (A + B)Q(HBT ) R↓ +(A + C)Q(HBT ) R↑ + (B + C)Q(HBT ) L↑ +AQ(HOM) R↓,R↑ + BQ(HOM) R↓,L↑ + CQ(HOM) R↑,L↑ i (19) where A = |λff|2λ2 pf, B = |λff|2|λpb|2, C = λ2 pf|λpb|2 (these can be tuned by modifying the QPC parameters [44, 47]). The Fermi sea [48,49], the HBT [12], and the HOM [13,14] noise contributions read respectively: Q(F S) =e2 π Z d¯tdξfξ (1 −fξ) (20) Q(HBT ) a = e2 2π Z d¯tdξ∆W(e) a (¯t, ξ) (1 −2fξ) (21) Q(HOM) a,b = −e2 π Z d¯tdξ∆W(e) a (¯t, ξ)∆Wb(¯t + δ, ξ) (22) with a and b the channels of injection and ∆W(e) a (¯t, ξ) = Z dτeiξτ∆G(e) a \u0010 ¯t + τ 2, ¯t −τ 2 \u0011 (23) the Wigner function [50] obtained as a partial Fourier transform of the excess first", + "Z d¯tdξ∆W(e) a (¯t, ξ)∆Wb(¯t + δ, ξ) (22) with a and b the channels of injection and ∆W(e) a (¯t, ξ) = Z dτeiξτ∆G(e) a \u0010 ¯t + τ 2, ¯t −τ 2 \u0011 (23) the Wigner function [50] obtained as a partial Fourier transform of the excess first order coherence [25] ∆G(e) a (t, t′) = ⟨Ψ † a(t′)Ψa(t)⟩ρ −⟨F|Ψ † a(t′)Ψa(t)|F⟩. (24) In the absence of spin-flip (A = B = 0), C becomes the product of the transmission and reflection probability of the QPC and one recovers what is observed in the IQH case. On a more general ground, when one of the scatter- ing amplitudes is zero only one of the A, B, C parameters survives and we recover an equivalent result to that of the IQH situation. In experiments, one subtracts the Fermi sea contribu- tions to define the excess noise: ∆Q = Q −(A + B + C)Q(F S). (25) When considering the emission of a pair of identical WPs in the form of Eq. (9) from a PES, the HBT contributions reduce to Q(HBT ) a ≈e2 while the HOM contributions [13] read Q(HOM) a,b (δt) ≈−2e2 exp (−Γ|δt|). Notice that, for the sake of simplicity, we have neglected the overlap of the injected electron WP with the Fermi distribution of the other channels (emission high above the Fermi sea and at very low temperature [12]). 3.2.1 Two-electron collision. We consider the injec- tion of electrons into the (R, ↑) and the (L, ↑) incoming channels. Here, only PES1 and PES3 (respectively green and orange circles in Fig. 3) are “on”. This process is there- fore the QSH equivalent of the IQH case (equal spin in- jection). The more relevant physical quantity to look at is the ratio between the HOM noise (two sources emitting to- gether with finite delay δt) and the sum of the HBT noises associated with the same sources: q(2) R↑,L↑(δt) ≈1 −Ie−Γ |δt| (26) where I = 2C/(A + B + 2C) is the visibility (see Fig. 4). Eq. (26) predicts a dip in the noise for electrons reaching the QPC with a delay such as Γ|δt| < 1. Moreover, the ex- ponential form of the dip is reminiscent of the WP profile. As in the IQH case discussed above, this Pauli dip is due to the fermionic statistics of the electrons [14], while the re- duced visibility (compared to [13]) is due to the presence of additional channels coupled at the QPC [51]. Indeed, more outgoing channels lead to more partitioning at the QPC and a consequent enhancement of the HBT noise contribution. This effect can be very small or conversely quite important depending on the visibility I (and consequently of the in- tensity of γsp and γsf) as shown in Fig. 5. Notice that, for γsf = 0 (absence of spin-flipping) one has I = 1 and we recover the result of the IQH. It is worth mentioning that the suppression of the visibility discussed here has a dif- ferent physical origin with respect to the one", + "as shown in Fig. 5. Notice that, for γsf = 0 (absence of spin-flipping) one has I = 1 and we recover the result of the IQH. It is worth mentioning that the suppression of the visibility discussed here has a dif- ferent physical origin with respect to the one observed in the IQH effect at filling factor ν = 2 [14,15]. Indeed, even if in this case several scattering channels are present, the QPC can be easily experimentally tuned in a region where this effect is absent (partial transmission of the outer chan- nel and total reflection of the others). Moreover, additional checks allows to unequivocally identify the inter-channel interaction as the dominant cause for the loss of contrast in the IQH case [52]. Due to spin flip processes occurring at the QPC, op- posite spin electron interferometry is also possible for this kind of device. For electrons of the same chirality we have: q(2) R↓,R↑(δt) ≈1 −J e−Γ |δt| (27) Copyright line will be provided by the publisher 6 Dario Ferraro et al.: Electronic quantum optics... γsp γsf γsp γsf γsp γsf -2 -1 0 1 2 -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1 Figure 4 (Color on-line) Density plot of I (left), J (middle) and K (right) as a function of γsp and γsf. Picture taken from [18]. δt q(2) R↑,L↑(δt) Figure 5 (Color on-line) Behavior of q(2) R↑,L↑(δt) as a func- tion of δt (in units of Γ −1) for different values of spin- flipping and spin-preserving amplitudes: γsp = 2, γsf = 0 (full black curve), γsp = 2, γsf = 1.5 (dashed green curve) and γsp = 1, γsf = 0.3 (dotted blue curve). Picture taken from [18]. when PES1 and PES2 are “on” (green and magenta cir- cles in Fig. 3). Notice that we have defined a new visibility factor J = 2A/(2A + B + C) (see Fig. 4). It is clear that the interference process between (R, ↑) and (L, ↑) electrons can be mapped into the one involving (R, ↑) and (R, ↓) electrons by exchanging the spin-preserving and spin-flipping contributions. Notice that at γsp = 0 (B = C = 0) we achieve the maximum visibility (J = 1). The present setup offers also the novel possibility to realize the interference of electrons with opposite spin and opposite chirality: q(2) R↓,L↑(δt) ≈1 −Ke−Γ |δt| (28) with PES2 and PES3 turned “on” (magenta and orange circles in Fig. 3) and K = 2B/(A + 2B + C). Maximum visibility is reached when A = C = 0, namely in a circle of radius 1 in the (γsp, γsf) plane (see Fig. 4). δt q(2) R↓,L↑(δt) q(2) R↓,R↑(δt) Figure 6 (Color on-line) Behavior of q(2) R↓,R↑(δt) (full black) and q(2) R↓,L↑(δt) (dashed green) as a function of the δt (in units of Γ −1). Parameters are γsp", + "0, namely in a circle of radius 1 in the (γsp, γsf) plane (see Fig. 4). δt q(2) R↓,L↑(δt) q(2) R↓,R↑(δt) Figure 6 (Color on-line) Behavior of q(2) R↓,R↑(δt) (full black) and q(2) R↓,L↑(δt) (dashed green) as a function of the δt (in units of Γ −1). Parameters are γsp = γsf = 2. Pic- ture taken from [18]. In the three different two-electron collision configura- tions (Eqs. (26), (27) and (28)) the maximal visibility oc- curs when one of the scattering amplitudes (respectively λff, λpb or λpf) is zero. Indeed, in this case, only two out- going channels are available for the two electrons and we recover a zero noise as in the IQH case. However, the noise suppression observed for collision of electrons with opposite spin is by far not trivial and is due to the constraints imposed by TRS and charge con- servation in the QSH system (see Ref. [53] for a simi- lar discussion in the case of a continuous current). This phenomenon, known as Z2 dip, has also been discussed in Ref. [40] for a specific range of parameters (γsp = 0, γsf ̸= 0), a particular case of the more general analysis reported here. 3.2.2 Three-electron collision. Unique to the QSH effect is the possibility to translate three-photon HOM ex- periments [54] to EQO. Here, the three PES of the setup in Fig. 3 are operating. The delays in the electron emis- sion are respectively δt1 between (R, ↓) and (R, ↑); δt2 between (R, ↓) and (L, ↑). Consequently, (δt2 −δt1) rep- Copyright line will be provided by the publisher pss header will be provided by the publisher 7 resents the delay between (R, ↑) and (L, ↑). As previously we obtain a normalized noise: q(3)(δt1, δt2) ≈1 − A A + B + C e−Γ |δt1| − B A + B + C e−Γ |δt2| − C A + B + C e−Γ |δt2−δt1|. (29) Remarkably enough, a perfect synchronization between the PES leads to q(3)(δt1 = 0, δt2 = 0) = 0 for an arbitrary QPC. This total noise suppression is a conse- quence of the interplay between the fermionic statistics and the TRS in the QSH systems. This can be understood by considering the input state a† R↑a† R↓a† L↑|F⟩(creation of three electrons simultaneously, on in each input channel). In terms of the scattering matrix, one can rewrite this as: a† R↑a† R↓a† L↑|F⟩= \u0010 λ∗ pbb† L↑+ λpfb† R↑+ λ∗ ffb† R↓ \u0011 × \u0010 λ∗ pbb† L↓+ λ∗ ffb† R↑+ λpfb† R↓ \u0011 × \u0010 λpfb† L↑+ λffb† L↓+ λ∗ pbb† R↑ \u0011 |F⟩. (30) TRS guarantees that each incoming operator a† ασ can be expressed as a linear combination of only 3 out of the 4 outgoing operators b† ασ. Therefore, exploiting the unitarity of the scattering matrix and the Pauli principle one obtains a† R↑a† R↓a† L↑|F⟩= \u0010 λpbb† L↑b† L↓+ λpfb† R↓b† L↑ +λffb† R↓b† L↓ \u0011 b† R↑|F⟩. (31) As it is easy to note, this leads to the superposition of three outgoing states, each involving", + "ασ. Therefore, exploiting the unitarity of the scattering matrix and the Pauli principle one obtains a† R↑a† R↓a† L↑|F⟩= \u0010 λpbb† L↑b† L↓+ λpfb† R↓b† L↑ +λffb† R↓b† L↓ \u0011 b† R↑|F⟩. (31) As it is easy to note, this leads to the superposition of three outgoing states, each involving the creation of an electron in the (R, ↑) outgoing channel. Consequently this channel is always populated (no current fluctuations) independently of the final outcome of the scattering process. The parti- tion noise in this channel vanishes, which constitutes a di- rect consequence of the interplay between TRS and Fermi statistics. In the absence of synchronized injections the phe- nomenology is even richer and crucially depends on the QPC parameters (γsp and γsf). Situations may oc- cur where the noise suppression is dominated by the (R, ↓|L, ↑) interference channel (see top panel of Fig. 7), or oppositely equal spin injection (R, ↑|L, ↑) can be fa- vored (see bottom panel of Fig. 7). According to this, the present setup allows to investigate different interference configurations, by changing the function of the QPC. Al- ternatively, HOM interferometry can be seen as a way to quantify the characteristics of the QPC, and in particular to assess the relative weight between the spin-preserving and the spin-flipping tunneling amplitudes. To summarize, 2D topological insulators exhibiting the QSH effect, are characterized by a very rich physics re- lated to the peculiar connection between spin and momen- tum of the electrons propagating along the edges. Equal δt1 δt1 δt2 δt2 Figure 7 (Color on-line) Density plot of q(3)(δt1, δt2) as a function of δt1 and δt2 (in units of Γ −1) for γsp = 1, γsf = 0.8 (top) and γsp = 2, γsf = 1.5 (bottom). Pic- ture taken from [18]. spin, as well as opposite spin interference are allowed here as a consequence of TRS. Moreover, three-electron injec- tion/interference phenomena similar to the ones observed for photons in the conventional quantum optics can be in- vestigated in this setup, differently from what happens in the IQH case [18]. 4 Non-local interference and Hong-Ou-Mandel collisions of individual Bogoliubov quasiparticles. EQO scenarios are now revisited using a SES in the IQH regime put in proximity with a superconductor (SC) (see Fig. 8). The electrons which are injected can perform sev- eral Andreev reflections [55] and can thus be converted partially or totally into holes. Collisions between two Bo- goliubov quasiparticles at the location of a QPC can be obtained in the framework of HOM interferometry. Ref. [22], considered this setup first, with electron “beams” (DC voltage imposed between the two opposite edges) impinging on the QPC rather than single quasiparticle ex- citations. It has been argued [21,22] that since Bogoliubov quasiparticle creation operators are related by a unitary transformation to their annihilation counterpart, these ex- citations qualify as Majorana fermionic excitations. This constitutes an alternative proposal for Majorana fermions [20] compared to their topological superconductor coun- terparts [56,57] which give rise to a zero bias anomaly Copyright line will be provided by the publisher 8 Dario Ferraro", + "unitary transformation to their annihilation counterpart, these ex- citations qualify as Majorana fermionic excitations. This constitutes an alternative proposal for Majorana fermions [20] compared to their topological superconductor coun- terparts [56,57] which give rise to a zero bias anomaly Copyright line will be provided by the publisher 8 Dario Ferraro et al.: Electronic quantum optics... in tunneling experiments [58,59]. The DC proposal [22] SES SC W Figure 8 (Color online) Schematic view of a single elec- tron source (SES) injecting electron and hole WPs into a IQH edge state at filling factor ν = 2 coupled with a SC contact of length W. Picture taken from [19]. failed to address the single shot creation and collision of two Bogoliubov quasiparticles, leading in principle to the annihilation of single Majorana excitations at the QPC. Moreover, this proposal requires the measurement of high frequency noise [60] in a normal metal/SC de- vice. More than a decade ago the finite frequency noise of a normal metal/SC junction was computed [61], but it has so far eluded experimental observation. Single par- ticle/quasiparticle injection, presents the great advantage that only zero frequency noise needs to be measured [4]. Here, the injection process is first characterized in terms of current and noise, exhibiting the non-conservation of the charge and the conservation of the excitation number. Next, EQO interferometric configurations with a QPC are studied, and it is shown that the averaged current (first order coherence [1]) is independent of the SC phase differ- ence, while oscillations dependent on the SC phase differ- ence appear in the noise (second order coherence [62,63]), which constitutes a clear signature of non local phenomena [19] together with the annihilation of Majorana excitations at the QPC. 4.1 Source of Bogoliubov quasiparticles. Two edge channels at ν = 2 (Zeeman splitting and inter-channel interaction are ignored [64]) are coupled to a SES and to a SC contact of length W (see Fig. 8). The SES injects into the channels an electron (a hole) with well defined WPs. A spin-singlet coupling between the Hall channels and the SC contact can for instance be realized in graphene [65, 66]. The action of the SC on incoming electrons with energy below the induced SC gap ∆s is described in terms of an energy dependent 4 × 4 transfer matrix M, constrained by unitarity and particle-hole symmetry [22,67]: M(ξ) = (τx ⊗I) M∗(−ξ) (τx ⊗I) . (32) In the above expression I is the identity in spin space, while from now on we indicate with τi (i = x, y, z) the Pauli matrices acting on the electron-hole space and with σi (i = x, y, z) the ones related to the spin degree of free- dom. According to this, the transfer matrix M is applied to a 4-component spinor state [22]: c(ξ) =      ce,↑(ξ) ce,↓(ξ) ch,↑(ξ) ch,↓(ξ)     , (33) where e (h) indicates the electron (hole) state and ↑(↓) the up (down) spin direction and ξ is the energy of the in- coming excitation. The particle-hole symmetry in", + "4-component spinor state [22]: c(ξ) =      ce,↑(ξ) ce,↓(ξ) ch,↑(ξ) ch,↓(ξ)     , (33) where e (h) indicates the electron (hole) state and ↑(↓) the up (down) spin direction and ξ is the energy of the in- coming excitation. The particle-hole symmetry in Eq. (32) leads to the constraint c(ξ) = τx ⊗I \u0002 c†(−ξ) \u0003T [57]. The parameters which enter the transfer matrix are: δ = W/vF the time required for the excitation to cross the SC region; α ≈W/ls (ls = ¯hv/∆s the proximity-induced coherence length) and β ≈W/lm with lm = (¯h/eB) 1 2 the magnetic length of the Hall system (B the magnetic field); φ the phase of the SC; γ and γ′ the relative phase shifts of electrons and holes in presence of the magnetic field [22, 67,68]. After some algebra, the explicit form of this transfer matrix which takes into account Andreev reflection reads [22]: M(ξ) = eiξδeiΓ τzU(˜θ, φ)eiΓ ′τz (34) where we introduced the phase shifts Γ = γ + Ω, Γ ′ = γ′ + Ω, with Ω= arctan \u0010 β tan p α2 + β2/ p α2 + β2 \u0011 and the matrix U(˜θ, φ) = exp h i˜θσy ⊗(τx cos φ + τy sin φ) i , (35) with ˜θ an angle such that sin ˜θ = α sin( p α2 + β2)/ p α2 + β2. (36) By comparing the expression for the upper critical magnetic field in a Type II SC in terms of the coherence length Bc = Φ0/(2πl2 s) and the definition of the mag- netic length lm, the condition ls ≪lm (and consequently α ≫β) must be enforced to preserve the superconductiv- ity. We consider now a SES injecting a spin up electron with the same WP as in Eq. (9) into the SC region. For the sake of simplicity we will investigate the zero tempera- ture case. The outgoing state from the SC is a Bogoliubov quasiparticle (a coherent electron/hole superposition with opposite spin): |B⟩= We|e, ↑⟩+ Wh|h, ↓⟩ = cos ˜θ|e, ↑⟩+ sin ˜θe−iΦ|h, ↓⟩ (37) with Φ = 2Γ −φ and |e, ↑⟩, |h, ↓⟩a notation for elec- tron/hole outgoing states from the SC region. Copyright line will be provided by the publisher pss header will be provided by the publisher 9 The averaged total current and particle density outgo- ing from the SC region are defined as: ⟨ϕ|I|ϕ⟩= −e⟨ϕ| : ˜Ψ †τz ˜Ψ : |ϕ⟩ (38) ⟨ϕ|ρ|ϕ⟩= ⟨ϕ| : ˜Ψ † ˜Ψ : |ϕ⟩ (39) where −e < 0 and we have omitted the time dependence for notational convenience. Note that, in the above expres- sions, the definition ˜Ψ(t) = (4π)1/2 R +∞ −∞dξe−iξtM(ξ)c(ξ) for the outgoing spinor is required to avoid double count- ing [22]. Applying Wick’s theorem and considering well local- ized WPs in the positive energy domain, the current re- duces to: ⟨ϕ|I(t)|ϕ⟩= −e cos(2˜θ)ϕ(t −δ)ϕ∗(t −δ). (40) The outgoing electronic current of Eq. (40) differs from the incoming one [1,50] ⟨ϕ|Iin(t)|ϕ⟩≡−eϕ(t)ϕ∗(t), by a time delay δ and by a factor cos(2˜θ),", + "[22]. Applying Wick’s theorem and considering well local- ized WPs in the positive energy domain, the current re- duces to: ⟨ϕ|I(t)|ϕ⟩= −e cos(2˜θ)ϕ(t −δ)ϕ∗(t −δ). (40) The outgoing electronic current of Eq. (40) differs from the incoming one [1,50] ⟨ϕ|Iin(t)|ϕ⟩≡−eϕ(t)ϕ∗(t), by a time delay δ and by a factor cos(2˜θ), which takes into account the conversion of electrons into holes via Andreev reflections. This is simply the difference between the prob- ability |We|2 = cos2 ˜θ for the incoming electron to emerge as an electron and |Wh|2 = sin2 ˜θ to be converted into a hole. For ˜θ = 0 (|We|2 = 1 and |Wh|2 = 0) the SC con- tact only induces a delay, while for ˜θ = π/2 (|We|2 = 0 and |Wh|2 = 1) the incoming electron is completely con- verted into a hole and a Cooper pair enters into the SC. More importantly, for ˜θ = π/4 (|We|2 = |Wh|2 = 1/2) the electron and hole contributions compensate and no av- eraged current flows out. Nevertheless, this zero averaged current still bears fluctuations. The charge outgoing from the SC contact, Q = Z +∞ −∞ dt⟨ϕ|I(t)|ϕ⟩= −e cos(2˜θ) (41) is not conserved as a consequence of the creation/destruction of Cooper pairs in the SC. Conversely, due to the unitarity of the scattering matrix, the outgoing excitation density is given by: ⟨ϕ|ρ(t)|ϕ⟩= ϕ(t −δ)ϕ∗(t −δ) = ⟨ϕ|ρin(t −δ)|ϕ⟩, (42) which implies a mere time delay δ with respect to the in- coming one. The prefactor is given by |We|2 +|Wh|2 = 1, which illustrates the conservation of the number of injected excitations: N = Z +∞ −∞ dt⟨ϕ|ρin(t −δ)|ϕ⟩= 1 . (43) Both the non-conservation of the charge and the con- servation of the excitation number are encoded in the Bogoliubov-de Gennes Hamiltonian. We also consider the current noise at the output of the SC contact: Ssource = Z +∞ −∞ dtdt′⟨ϕ|I(t)I(t′)|ϕ⟩c = e2 sin2(2˜θ). (44) The above quantity is proportional to Pe·Ph: it vanishes in the absence of a SC contact (˜θ = 0), as expected [4,14], and when the incoming electron is completely converted into a hole (˜θ = π/2). Its maximum is reached for ˜θ = π/4, when the outgoing averaged current is zero. 4.2 Cross-correlated noise in a QPC geometry. We now investigate the outgoing cross-correlated noise in a QPC geometry where one or two SES (SES1 and SES2) inject electronic WPs with spin up in the vicinity of one or two SC regions (see Fig. 9). Σ SES1 SES2 SC1 SC2 c1 M1c1 c2 M2c2 a1 a2 I out 1 (t) I out 2 (t) Figure 9 (Color online) QPC geometry for Bogoliubov quasiparticles. The individual electron sources SES1 and SES2 inject electrons (described by the operators cj (j = 1, 2)), into the two SC contacts SC1 and SC2. The out- going excitation (Mjcj) reach the QPC and are parti- tioned according to the scattering matrix Σ. We consider the cross-correlated noise between the outgoing currents I out j (t) (written in terms of the operators aj). Picture taken from [19]. The annihilation", + "two SC contacts SC1 and SC2. The out- going excitation (Mjcj) reach the QPC and are parti- tioned according to the scattering matrix Σ. We consider the cross-correlated noise between the outgoing currents I out j (t) (written in terms of the operators aj). Picture taken from [19]. The annihilation spinors outgoing from the QPC are related to the ones emitted by the two SES through the scattering matrix, which is parametrized by transmission (reflection) coefficients T (R). The zero frequency cross- correlated noise outgoing from the QPC is again given by Eq. (1). 4.2.1 Hanbury-Brown-Twiss contribution. When only one of the two SES (labeled j) is “on” we obtain the HBT contribution to the noise. At zero temperature, the injected excitations crossing the SC contact are converted into Bogoliubov quasiparticles which reach the QPC and get partitioned [12]. This contribution to the noise is: S(j) HBT = −e2RT cos2(2˜θj). (45) This represents the shot noise associated with a WP carrying charge Q (see Eq. (41)) and is therefore propor- tional to Q2. In the absence of SC (˜θ = 0), SHBT = Copyright line will be provided by the publisher 10 Dario Ferraro et al.: Electronic quantum optics... −e2RT as expected (see Eq. (7) in the zero temperature limit). For ˜θj = π/4, the state which reaches the QPC (a balanced superposition of electrons and holes) gener- ates no noise at all. This is because this individual zero charged quasiparticle excitation presents a non trivial in- ternal structure which cannot be simply described in terms of an incoherent mixture of electrons and holes [69,70]. For ˜θj = π/2, the electron is completely converted into a hole, and the noise is the same as for ˜θj = 0. 4.2.2 Hong-Ou-Mandel contribution. If both the SES are “on” we obtain the HOM noise signal [13,14] SHOM = ∆SHOM + S(1) HBT + S(2) HBT (46) with ∆SHOM/S0 = A(δ1 −δ2 −η) h 1 + cos(2˜θ1) cos(2˜θ2) −cos(Φ12) sin(2˜θ1) sin(2˜θ2) i , (47) S0 = e2RT , η the time delay in the emission between the two SES, Φjk = 2Γj −2Γk −φj + φk, and the squared overlap A(τ) = R +∞ −∞dtϕ∗(t −τ)ϕ(t) 2 between identi- cal WPs with a delay τ. This constitutes a general, central analytical result, as it addresses the HOM collision of two unsynchronized Bogoliubov quasiparticles. When the two SC regions only differ in their order pa- rameter phase and the two SES are properly synchronized (η = 0) one obtains the simplified expression: S2SC HOM = e2RT sin2(2˜θ) [1 −cos(φ1 −φ2)] (48) which clearly shows a non-local dependence on the differ- ence of the SC order parameter phases as already pointed out in Ref. [22] in the DC regime. The device shows no dependence on the SC phase at the level of the averaged current (first order coherence), but presents an oscillatory modulation in the noise (second order coherence). This is a clear demonstration of the fact that noise measure- ments in this kind of devices allow to access purely two- quasiparticle effect in analogy to what", + "phase at the level of the averaged current (first order coherence), but presents an oscillatory modulation in the noise (second order coherence). This is a clear demonstration of the fact that noise measure- ments in this kind of devices allow to access purely two- quasiparticle effect in analogy to what was discussed in the framework of IQH effect for the interferometers of Refs.[71,72] or the revisitation of the Franson interferom- eter [73] proposed in Refs. [27,63]. If in Eq. (48) φ1 −φ2 ̸= 0 (mod. 2π), the noise van- ishes only when two electrons or two holes reach the QPC at the same time (˜θ = 0 or ˜θ = π/2) as a consequence of the Pauli principle [13]. Remarkably, the noise reaches its maximum for ˜θ = π/4. To explain this, look at the struc- ture of the ∆SHOM term in Eq. (47). By considering two Bogoliubov excitations in the form of Eq. (37) simultane- ously reaching the QPC, this contribution to the noise is proportional to: |W1 e W2 e ∗−W1 hW2 h ∗|2 = | cos ˜θ1 cos ˜θ2 −sin ˜θ1 sin ˜θ2e−i(Φ1−Φ2)|2. (49) It thus corresponds to the difference between the product of electron and hole probability amplitudes. In particular for ˜θ1 = ˜θ2 = π/4 the Bogoliubov quasiparticles carry zero charge and zero shot noise, but are by far not trivial excitations with a complex structure given by the coherent superposition of electrons and holes which can be detected only at the level of the two quasiparticle interferometry. The above argument is also useful in order to understand why the HOM contribution to the noise in Eq. (48) is zero for φ1−φ2 = 0 (mod. 2π). Under this condition indeed, the ∆SHOM term exactly compensates the two (equal) SHBT contributions in analogy to what occurs for the IQH case in absence of interaction. The peculiar structure of the HOM noise contribution for two synchronized Bogoliubov excitations directly re- flects into the divergences associated with the ratio: R2SC = S2SC HOM S(1) HBT + S(2) HBT = −1 2 tan2(2˜θ) [1 −cos(φ1 −φ2)] . (50) We can also achieve collisions between single electrons and Bogoliubov quasiparticles: it is now shown that the electron colliding with the Bogoliubov quasiparticle allows to probe the content of the latter. Starting from Eq. (47) the corresponding noise becomes: S1SC HOM = e2RT nh 1 + cos(2˜θ) i A(δ1 −η) −cos2(2˜θ) −1 o (51) with a maximum WP overlap A = 1, the reference elec- tron interferes with: a) another electron (˜θ = 0) leading to a zero noise (Pauli principle); b) with a hole (˜θ = π/2) with a consequent minimum of the noise [13]; c) a more general Bogoliubov quasiparticle. In the latter case the cross-correlated noise assumes positive values when the electron component of the Bogoliubov excitation domi- nates over the hole one (|We|2 > |We|2 and consequently for 0 < ˜θ < π/4). By decreasing the WP overlap, the ∆SHOM contribution to the noise is suppressed. However, even away from perfect synchronization, it is possible to observe positive and", + "the electron component of the Bogoliubov excitation domi- nates over the hole one (|We|2 > |We|2 and consequently for 0 < ˜θ < π/4). By decreasing the WP overlap, the ∆SHOM contribution to the noise is suppressed. However, even away from perfect synchronization, it is possible to observe positive and negative regions from which we can extract the dominant contribution to the Bogoliubov quasi- particle. The situation in the case of a WP exponential in time (see Eq. (9)), where A(τ) = e−Γ |τ|, is illustrated by the density plot in the upper panel of Fig. 10. This represents an extremely useful tool to extract in- formation about the structure of the Bogoliubov excitations through interferometric experiments with a known source (electronic WP). These observations indicate that the con- sidered setup offers richer possibilities to implement a to- mographic protocol by means of HOM interferometry with respect to what was proposed in the electronic case [74]. Copyright line will be provided by the publisher pss header will be provided by the publisher 11 -1.5 -1 -0.5 0 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 π 4 π 4 π 2 π 2 ˜θ ˜θ δ1 −η Figure 10 (Color online). Upper panel. Density plot of S1SC HOM in units of S0 = e2RT . “Blue” identifies nega- tive regions where the hole contribution dominates over the electron one, while the little red area represents the positive noise. Lower panel. Density plot of R1SC as a function of δ1 −η and ˜θ. The blue area corresponds to negative val- ues of the ratio which cannot be reached in conventional electronic quantum optics experiments. Picture taken from [19]. Another relevant quantity to explore is given by the ratio: R1SC = S1SC HOM/(S(1) HBT + S(2) HBT ) = 1 −1 + cos(2˜θ) 1 + cos2(2˜θ) A(δ1 −η). (52) This also contains negative regions (blue areas in the lower panel of Fig. 10) which are forbidden in conventional elec- tron quantum optics situations (˜θ = 0) [13] due to the con- straints imposed by the charge conservation. 5 Conclusions. Scattering theory thus allows to study a number of EQO interferometric situations. In IQH regime, theory has already been confronted with experi- ment [14]. The prediction of the HOM dip is solid, but its visibility does not correspond to what was expected [13]. Experiments cannot easily be achieved at filling factor one and when several channels are present, interactions be- tween them lead to charge fractionalization thus decoher- ence effects accompanied by a reduced visibility [15]. Two new directions for EQO have been proposed: the study of HOM collisions in the QSH effect in 2D topological insulators [18] and the possibility of observing non-local phenomena when colliding two Bogoliubov quasiparticles at the location of a QPC [19]. In these two setups, finite temperature effects were dis- carded. This should not change dramatically our predic- tions as long as the electrons WPs have a well defined en- ergy above the Fermi sea and, for the former case, well be- low the gap. However", + "location of a QPC [19]. In these two setups, finite temperature effects were dis- carded. This should not change dramatically our predic- tions as long as the electrons WPs have a well defined en- ergy above the Fermi sea and, for the former case, well be- low the gap. However they are likely to be affected when considering electron-hole interferometry [13] where the overlap between electron and hole WPs within an energy window set by the temperature is crucial. Further work should however address the issue of inter- channel interactions as in Ref. [15]. For the QSH setup, one suspects that Coulomb interactions between the counter- propagating edges could lead to a further decreasing in the visibilities of the Pauli and the Z2 dips in the two elec- tron collisions, and prevent a maximal visibility for the three electron dip. For Ref. [19], the approach of Ref. [15] could be directly transposed in order to predict the modi- fication of the Bogoliubov quasiparticle impinging on the QPC, and the modification of the HOM signal. 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Copyright line will be provided by the publisher", + "Tuning the electronic structures and transport properties of zigzag blue phosphorene nanoribbons Yipeng An,*,†,‡ Songqiang Sun,† Mengjun Zhang,† Jutao Jiao,† Dapeng Wu,§ Tianxing Wang ‡ and Kun Wangǁ †College of Physics and Materials Science & International United Henan Key Laboratory of Boron Chemistry and Advanced Energy Materials, Henan Normal University, Xinxiang 453007, China E-mail: ypan@htu.edu.cn ‡Department of Physics and Astronomy, University of California, Irvine, California 92697, USA §School of Chemistry and Chemical Engineering & International United Henan Key Laboratory of Boron Chemistry and Advanced Energy Materials, Henan Normal University, Xinxiang 453007, China ǁDepartment of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA Abstract—In recent years, single element two-dimensional (2D) atom crystal materials have aroused extensive interest in many applications. Blue phosphorus, successfully synthesized on Au (111) substrate by molecular beam epitaxy not long ago, shows unusual geometrical and electronic structures. We investigate the electronic structures and transport properties of zigzag blue phosphorene nanoribbons (ZBPNRs) by using a first-principles method, which can be obviously tuned via different groups (i.e., -H, -O, and -OH) passivation on the both edges. The ZBPNRs-H and ZBPNRs-OH present a wide gap semiconductor property. While the ZBPNRs-O are metallic. Interestingly, the current–voltage (I-V) curves of ZBPNRs-O show a negative differential resistive (NDR) effect, which is independent on the ribbon width. The electric current through the ZBPNRs-O is mainly flowing along the both outside zigzag phosphorus chains via the way of P-P bond current. Through modifying the both edges with various functional groups, the ZBPNRs can display some important functional characteristics and become a candidate of NDR devices. I. INTRODUCTION VER the past decades, some two-dimensional materials, such as graphene [1]–[6], hexagonal boron nitride (h-BN) [7]–[9], silicon carbide (SiC) [10], [11], transition-metal dichalcogenides [12]–[15], phosphorene [16]–[18], and phosphorene-like structures [19], [20], have caused extensive interest, not only from a fundamental point of view to understand their intrinsic properties but also from a practical point of view to study their potential applications in future nano-devices. Phosphorene has a variety of allotropes based on their different atomic structure arrangement. It includes the common white, red, violet, and black allotropes, with the color defined by the bandgap. Phosphorene is studied with the hope of overcoming graphene’s deficiencies, such as its zero bandgap. It has been emerged as a promising new two-dimensional material due to its intrinsic and tunable bandgap, high carrier mobility and remarkable in-plane anisotropic electrical, optical and phonon properties [21], [22]. With its excellent transport properties, phosphorene has attracted a remarkable interest for potential applications in high-performance thin-film electronics, mid- and near-infrared optoelectronics, and for the development of conceptually novel devices that utilize its anisotropic properties [23], [24]. For instance, the Zhang [16] group synthesized a few layers of black phosphorus in an experiment using the scotch tape-based mechanical exfoliation method. They peeled thin flakes from bulk crystals and fabricated the field-effect transistors based on few layer black phosphorus crystals with a thickness down to a few nanometers. Due to the strong anisotropic atomic structure of black phosphorus, its electronic conductivity and optical response are sensitive to the magnitude", + "exfoliation method. They peeled thin flakes from bulk crystals and fabricated the field-effect transistors based on few layer black phosphorus crystals with a thickness down to a few nanometers. Due to the strong anisotropic atomic structure of black phosphorus, its electronic conductivity and optical response are sensitive to the magnitude and the orientation of the applied strain, which appears as a promising method to design novel photovoltaic devices that capture a broad range of solar spectrum [25]. Xie et al. [26] investigated the electronic and transport properties of zigzag phosphorene nanoribbons (ZPNRs) by performing first-principles calculations. It presents a negative differential resistance behavior, and the peak-to-valley current ratio is up to 100 under low biases. Zhu et al. [27] first predicted the existence of a previously unknown phase of O phosphorus, referred to as blue phosphorus. The blue phosphorus is hexagonal lattice crystals with a small out-plane distance of two sublattice sites, which is nearly as stable as black phosphorus. Lately, Chen et al. [28] synthesized single layer blue phosphorus on Au (111) substrate by molecular beam epitaxy for the first time. And the first-principles study shows that the nanoribbon edges will be significantly Fig. 1. The geometric and electronic structures of ZBPNRs. (a) - (c) Top and side views of the 5-ZBPNR-H, 5-ZBPNR-O, and 5-ZBPNR-OH unit cell, respectively. (d) - (f) Band structures of the 5-ZBPNR-H, 5-ZBPNR-O, and 5-ZBPNR-OH, respectively. changed under strain which results in a linear decrease of the gap of blue phosphorus nanoribbons with deformation aggravation [29]. In this work, we investigate the electronic structures and transport properties of zigzag blue phosphorene nanoribbons by a first-principles method, which can be obviously tuned through three different functional groups (i.e., -H, -O, and -OH) passivation on both edges. It makes them display semi-conductive and metallic characteristics, respectively. This edge-passivation way makes the blue phosphorous nanoribbons present more abundant electronic transport phenomena and can be useful for the design of phosphorous-based nano-electronic devices. II. COMPUTATIONAL METHODS We study the electronic transport properties of the ZBPNRs-based devices by density functional theory combined with the non-equilibrium Green’s functions method (NEGF-DFT) [30]–[33]. The GGA-PBE is adopted as the exchange correlation functional [34], [35]. The wave-functions of valence electrons are expanded in double-zeta plus polarization (DZP) basis set. The Hamiltonian and electronic densities are calculated in a real space grid defined with a plane-wave cut-off 150 Ry. The atomic structures are optimized until the absolute value of force acting on each atom is less than 0.01 eV/Å. The k-points grid 1 × 1 × 100 is used to sample the Brillouin zone of electrodes in the X, Y, and Z directions (transport direction), respectively. Fig. 2. The two-probe device model for the zigzag blue phosphorene nanoribbons. III. RESULTS AND DISCUSSION Up on structural optimization of the unpassivated ZBPNRs, the interior part of the nanoribbons experiences negligible structural change, but the edges of the nanoribbons show some degree of deformations [26]. They are not stable and easily twisty like graphene nanoribbons, due to the strong dangling bond at the both edges. Therefore, we try to stabilize", + "the unpassivated ZBPNRs, the interior part of the nanoribbons experiences negligible structural change, but the edges of the nanoribbons show some degree of deformations [26]. They are not stable and easily twisty like graphene nanoribbons, due to the strong dangling bond at the both edges. Therefore, we try to stabilize the ZBPNRs by passivating the both edges with -H, -O, and -OH groups. Fig. 1(a)-(c) shows the unit cells of zigzag blue phosphorene nanoribbons we studied, whose both edges are passivated with -H, -O, and -OH functional groups, respectively. We label the width of ZBPNRs with the number of parallel zigzag chains. The ZBPNRs with n chains is thus named as n-ZBPNRs. To evaluate the structural stability of ZBPNRs with different groups of -H, -O, and -OH, we calculated their binding energy Eb, defined as ‒(Et ‒ EBP ‒ 2*Ea)/2, where Et is the total energy of ZBPNRs passivated with groups, EBP is the energy of unpassivated ZBPNRs, and Ea represents the energy of an isolated group. The results demonstrate that the ZBPNRs-O have the largest Eb and are the most stable structures, while the ZBPNRs-H have the smallest one. For example, the Eb of the 5-ZBPNR-H, 5-ZBPNR-O and 5-ZBPNR-OH is 4.3, 6.6 and 4.4 eV, respectively. These -H, -O, and -OH functional groups obviously tune the electronic structures of ZBPNRs and arouse them displaying distinguishable band structures. For example, the ZBPNRs-H present a wide gap semiconductor property, and their indirect band gap (~2.0 eV) changes little with increasing the ribbon width. It is different from the edge-passivated ZPNRs, which have a direct bandgap (~1.4 eV) [36], [37]. When the both edges of ZBPNRs are passivated with -O functional group, they show a metallic characteristic with two nearly degenerate bands (i.e. the band I and II dominated by O atoms, see Fig. 1(e)) crossing the Fermi level (EF). The ZBPNRs-O are metallic and same to the edge-oxidized ZPNRs. It is because oxygen atoms form weak unsaturated bonds with the pz orbital of P atoms and bring some new edge states within the band gap. These edge states of ZBPNRs-O present around the EF within the band gap, closing up the band gap of ZBPNRs-O [38]. Moreover, when the both edges of ZBPNRs are passivated with -OH group, they show a similar semiconductor property to the ZBPNRs-H, but with a direct band gap (~2 eV) which is independent on the ribbons width. Monolayer TMDs were found to be a direct band gap semiconductor (1.0−2.0 eV) and are thus favorite for the logic transistor applications. The ZBPNRs-H and ZBPNRs-OH have the comparable band gaps and may also have potential applications in the field of logic transistors [39], [40]. Fig. 3. (a) The transmission spectra of 5-ZBPNRs with different functional groups at the bias of 0 V. (b) The I-V curves of the ZBPNRs-O with different widths. It can obviously tune the electronic structures of ZBPNRs through passivating the both edges with such common functional groups. To further understand the effects of functional groups modifying the edges, we investigate the electronic transport properties of ZBPNRs.", + "0 V. (b) The I-V curves of the ZBPNRs-O with different widths. It can obviously tune the electronic structures of ZBPNRs through passivating the both edges with such common functional groups. To further understand the effects of functional groups modifying the edges, we investigate the electronic transport properties of ZBPNRs. As shown in Fig. 2, we build a two-probe device model of the zigzag blue phosphorene nanoribbons. It is composed of three parts: the central scattering region (C), which is about 25 Å in length, the left (L) and the right (R) electrodes. Both the left and the right electrodes are consisted of three repeated unit cells and applied periodic boundary condition. According to the first-principles calculation results, the semiconductor ZBPNRs-H and ZBPNRs-OH indeed present a current forbidden behavior. While, the metallic ZBPNRs-O are conductive. As shown in Fig. 3(a), the electron transmission spectra of the 5-ZBPNR-H and 5-ZBPNR-OH are similar, and they both present a larger gap near the EF, which suggests that little electric currents can propagate through these nanoribbons under the limited bias voltages. However, the 5-ZBPNR-O has a larger transmission coefficient near the EF, displaying good electroconductibility (see Fig. 3(a)). And its transmission spectrum displays a remarkable quantized characteristic. It is mainly due to the distinctive band structures of the 5-ZBPNR-O (see Fig. 1(e)). Fig. 4. The band structures for the L and R electrodes, and transmission spectra under the bias of 0.3 V (a) and 1.0 V (b) for the 5-ZBPNR-O. The shadows denote the bias window. The current-voltage characteristic is one of the main parameters to evaluate the performance of low-dimensional nano-devices. To obtain the I-V curves of these ZBPNRs-based materials, we applied the external bias voltage Vb (Vb = VL – VR, and VL/R is the bias voltage applied on the left/right electrode) to the two-probe device model. We use the Landauer–Büttiker formula [41] to describe the generated electric current I, b b L L R R 2 ( ) ( , )[ ( ) ( )]d e I V T E V f E f E E h    − = − − −  (1) More details of the calculation method can be found in the previous reports [30]–[32]. In order to further understand the electronic transport properties of ZBPNRs-O under limited bias voltages, the I–V curves of the ZBPNRs-O with various ribbon widths are calculated and depicted in Fig. 3(b). One can see that the ZBPNRs-O show the similar electronic transport characteristics, and their I–V curves overlap with each other, showing little dependency on the ribbon width. What’s more, the I–V curves display a NDR effect, and the currents reach the maximum value at the bias of 0.3 V. It suggests that the ZBPNRs-O can be used as a candidate of NDR nano-devices, and have the potential applications in nano-electronics, e.g., high-frequency oscillators, multipliers, logic, and analog-to-digital converters. In general, for one-dimensional nanoribbons, their electronic structures (i.e., band structures) dominant the electronic transport behaviors. The NDR effect of the ZBPNRs-O can be understood most directly by the transmission spectra and", + "of NDR nano-devices, and have the potential applications in nano-electronics, e.g., high-frequency oscillators, multipliers, logic, and analog-to-digital converters. In general, for one-dimensional nanoribbons, their electronic structures (i.e., band structures) dominant the electronic transport behaviors. The NDR effect of the ZBPNRs-O can be understood most directly by the transmission spectra and band structures of the left and right electrodes under the biases of 0.3 and 1.0 V (see Fig. 4). The bands of the L and R electrodes shift downward and upward under a positive applied bias. The L and R electrode regions have an equilibrium electron distribution with their chemical potentials μL and μR, related to the applied bias voltage, μL ̶ μR = eVb. The electrons with energies in the bias window (BW) μL≤ E ≤ μR, give rise to a steady state electric current from the left electrode to the right one. With the increase of bias window, the overlapping area between the I/II and Iʹ/IIʹ bands of the electrodes is reduced, and the transmission coefficient decreases with the decrease of Fig. 5. (a) and (b) show the transmission eigenstates of the 5-ZBPNR-O under the bias of 0.3 V. (c) and (d) show the top and side views of the transmission pathways under 0.3 V bias, respectively. The isovalue of transmission eigenstates is set to 0.44 Å(-3/2). band overlaps. As shown in Fig. 4(a), the energy band of the left electrode has the most overlap with the right electrode at 0.3 V. The electron transmission coefficient is about 4, and the current reaches the peak value (about 25µA). However, when the bias voltage is increased to 1.0 V, the overlaps between the left and the right electrode bands decrease a little (see Fig. 4(b)), which results in a significant decrease in the electron transmission coefficient in the BW. Correspondingly, the electric current is reduced to about 9 μA. And then the NDR phenomenon is appeared subsequently (see Fig. 3(b)). To understand the intuitive electronic transmission mechanism of the ZBPNRs-O, we further analyzed the transmission eigenstates and transmission pathways (i.e. local currents) [42] at the transmission peak (E = 0 eV) under the bias of 0.3 V, as shown in Fig. 5 and 6. For the odd chains, taking the 5-ZBPNR-O as an example, there are two degenerate transmission eigenstates (see Fig. 5(a) and (b)), which are mainly located at one side of the nanoribbons. And the transmission eigenstates are mostly contributed by the 3p orbitals of phosphorus atoms. Transmission pathway is an analysis option which splits the transmission coefficient into local bond contributions, Tij. If the transmission pathways of the system is divided into two parts (A, B), the total transmission coefficient is obtained through the sum of pathways A and B , ( )= ( ) ij i A j B T E T E    (2) The arrows refer to the direction of electric current. As shown in Fig. 5(c) and (d), for the 5-ZBPNRs-O, there are two parallel transmission pathways, which are located at the both edges of nanoribbons. Moreover, one can see that the", + "j B T E T E    (2) The arrows refer to the direction of electric current. As shown in Fig. 5(c) and (d), for the 5-ZBPNRs-O, there are two parallel transmission pathways, which are located at the both edges of nanoribbons. Moreover, one can see that the local currents are completely composed of the P-P bond current type (i.e., electric current via the P-P chemical bond). It is noted that, the outside oxygen atoms give some contributions to the transmission eigenstates. However, they are too localized and give little contribution to the electron transmission. Interestingly, the transmission eigenstates of the ZBPNRs-O show an odd-even effect. The transport eigenstates of the ZBPNRs-O with odd chains are only Fig. 6. (a) and (b) show the transmission eigenstates of the 6-ZBPNR-O under the bias of 0.3 V. (c) and (d) show the top and side views of the transmission pathways at 0.3 V bias, respectively. The isovalue of transmission eigenstates is set to 0.31 Å(-3/2). distributed on one side of the nanoribbons. For the ZBPNRs-O with even chains, however, taking the 6-ZBPNR-O as an example, it has two degenerate transmission eigenstates (see Fig. 6(a) and (b)), which are mainly located at the phosphorous atomic chains of both edges of nanoribbons. Moreover, the transmission channels of ZBPNRs-O with even chains has two transport pathways on the phosphorous atomic chains of both edges of nanoribbons. And they are symmetrical relative to the Z axis rather than parallel. The electric currents flowing through the 6-ZBPNR-O are mostly composed of the P-P bond current. IV. CONCLUSIONS In this paper, the electronic transport properties of zigzag blue phosphorene nanoribbons are investigated using first-principles method. Our calculations results demonstrate that, it can be tuned obviously through passivating the both edges of nanoribbons with some functional groups (i.e., -H, -O, and -OH). The ZBPNRs-H present a wide gap semiconductor property, and their indirect band gap changes little with increasing the ribbon width. While, the ZBPNRs-O display the metallic characteristics, and their I-V curves show an interesting NDR effect, which are independent on their ribbon widths. 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Kotosonov Institute of Carbon, Moscow, Russia Abstract. The electron-electron interaction in carbon nanostructures was studied. A new method which allows to determine the electron-electron interaction constant λc from the analysis of quantum correction to the magnetic susceptibility and the magnetoresistance was developed. Three types of carbon materials: arc-produced multiwalled carbon nanotubes (arc-MWNTs), CVD-produced catalytic multiwalled carbon nanotubes (c-MWNTs) and pyrolytic carbon were used for investi- gation. We found that λc=0.2 for arc-MWNTs (before and after bromination treatment); λc = 0.1 for pyrolytic graphite; λc > 0 for c-MWNTs. We conclude that the curvature of graphene layers in carbon nanostructures leads to the increase of the electron-electron interaction constant λc. Key words: Electron-electron interaction; Nanostructures; Electronic transport; Galvanomagnetic effects; Quantum localization 1. Introduction The carbon nanostructures are formed of graphene layers which always have some curvature. As a result, these materials are characterized by new properties which are not present in graphite consists of plane graphene layers. The curvature of the graphene layers influences the electron-electron interaction in these systems. The most interesting consequence of the graphene layers curvature is the existence of a superconducting state in bundles of single-walled carbon nanotubes with diameters of 10 Å at temperatures below Tc ≈1 K (Kociak et al., 2001) as well as the onset of superconductivity at Tc ≈16 K in nanotubes with diameters of 4 Å (Tang et al., 2003) and at Tc ≈12 K in entirely end-bonded multiwalled carbon nanotubes (Takesue et al., 2006). In contrast, in graphite no superconducting state is observed. Y. Kopelevich et al. (Kopelevich et al., 2000) proposed that supercon- ductivity may appear in ideal graphite and that the absence of superconductivity c⃝2018 Springer. Printed in the Netherlands. 2 A.I. ROMANENKO, O.B. ANIKEEVA, T.I. BURYAKOV... in real samples is related to the defects always present in graphite. According to theoretical predictions of (Gonzalez et al., 2001) topological disorder can lead to an increase in the density of states at the Fermi surface and to an instability of an electronic subsystem. These changes in the electronic system could lead to a superconducting state. However, such topological disorder leads to a curvature of initially flat graphene layers. We assume, therefore, that in carbon nanostructures the superconducting state is related to the curvature of graphene layers. The curva- ture of surfaces is always present in the crystal structure of nanocrystallites. As a result, in such structures the electron-electron interaction should be modified. This paper is devoted to the analysis of experimental data which allows to determine the electron-electron interaction constant λc in carbon nanostructures formed by curved graphene layers. 2. Experimental methods The method of our investigation of the electron-electron interaction constant is based on the joint analysis of quantum corrections to the electrical conductance, magnetoconductance and magnetic susceptibility. For all", + "data which allows to determine the electron-electron interaction constant λc in carbon nanostructures formed by curved graphene layers. 2. Experimental methods The method of our investigation of the electron-electron interaction constant is based on the joint analysis of quantum corrections to the electrical conductance, magnetoconductance and magnetic susceptibility. For all nanostructures formed by graphene layers the presence of structural defects leads to the diffusive motion of charge carriers. As a result, at low temperatures, quantum corrections to the electronic kinetic and thermodynamic quantities are observed. For the one-particle processes (weak localization - WL (Kawabata, 1980; Lee and Ramakrishnan, 1985)) these corrections arise from an interference of electron wave functions propagating along closed trajectories in opposite directions, provided the lengths l of these trajectories are less then the phase coherence lengths Lϕ(T) = (Dτϕ)1/2 (D is the diffusion constant and τϕ = T −p is the characteristic time for the loss of phase coherence with an exponent p = 1 ÷ 2). As a result, the total conductance of the system is decreased. Lϕ(T) increases with decreasing temperature which, in turn, leads to the decrease of the total conductance. In a magnetic field there is an additional contribution to the electronic phase, which has an opposite sign for op- posite directions of propagation along the closed trajectory. As a result, the phase coherence length is suppressed: LB = (ℏc/2eB)1/2 < Lϕ. Here LB = (ℏc/2eB)1/2 the magnetic length, c is the light velocity, e - the electron charge, B - the magnetic field. This leads to negative magnetoresistance, i.e. to an increase of conductance in a magnetic field. Quantum corrections also arise from the interaction between electrons (interaction effects - IE (Al’tshuler et al., 1983)). These corrections arise due to the phase memory between two consecutive events of electron-electron scattering. If the second scattering event happens at a distance shorter than the coherence length, LIE = (Dℏ/kBT)1/2, from the first one (LIE being the length on which the information about the changes of the electronic phases due to the first scattering event is not yet lost), the second scattering will depend on the first one. As a result the effective density of states on the Fermi surface vF is ELECTRON-ELECTRON INTERACTION IN CARBON NANOSTRUCTURES 3 renormalized. Interaction effects contribute not only to electrical conductance, but also to thermodynamic quantities depending on vF - magnetic susceptibility χ and heat capacity C. 3. Results and discussion 3.1. ARC-PRODUCED MULTIWALLED CARBON NANOTUBES A characteristic peculiarity of our arc-MWNTs (Okotrub et al., 2001; Romanenko et al., PSS, 2002) is the preferential orientation of the bundles of nanotubes in the plane perpendicular to the electrical arc axis. The volume samples of our arc- MWNTs show anisotropy in their electrical conductivity σII/σ⊥≈100 (Okotrub et al., 2001; Romanenko et al., PSS, 2002). σII is the electrical conductivity in the plane of preferential orientation of the bundles of nanotubes, σ⊥is the conduc- tance perpendicular to this plane. The average diameter of individual nanotubes is dMWNT ≈140 Å. According to the electron paramagnetic resonance data, the concentration of paramagnetic impurities in our", + "PSS, 2002). σII is the electrical conductivity in the plane of preferential orientation of the bundles of nanotubes, σ⊥is the conduc- tance perpendicular to this plane. The average diameter of individual nanotubes is dMWNT ≈140 Å. According to the electron paramagnetic resonance data, the concentration of paramagnetic impurities in our samples is less than 10−6. This excludes a substantial contribution of the impurities to the susceptibility. The MWNTs brominated at room temperature in bromine vapour (Romanenko et al., PSS, 2002) have a composition of CBr0.06. The addition of bromine leads to an increase of the conductivity, which can be attributed to an increase in the concentration of hole current carriers. According to experimental and theoretical data, the basic contribution in χ of quasi-two-dimensional graphite (QTDG), including MWNTs, gives orbital mag- netic susceptibility χor connected with extrinsic carriers (EC). Figures 1(a) and 2(a) present the magnetic susceptibility χ of arc-MWNTs samples before bromi- nation and after bromination as a function of temperature respectively. Available models well reproduce the temperature dependence of magnetics susceptibility for MWNTs only at T > 50 K. In the low-temperature region the experimental data deviate from the theoretical ones. According to theoretical consideration the magnetic susceptibility χ of quasi-two-dimensional graphite (QTDG) is generally contributed by the component χD (Kotosonov et al., 1997) χor(T) = − 5.45 × 10−3γ2 0 (T + δ)[2 + exp(η) + exp(−η)], (1) where γ0 is the band parameter for two-dimensional case, δ is the additional temperature formally taking into account ”smearing” the density of states due to electron nonthermal scattering by structure defects, η = EF/kB(T + δ) represents reduced Fermi level (EF), kB is the Boltzmann constant. Using an electrical neu- trality equation in the 2D graphite model (Kotosonov et al., 1997) η can be derived by η = sgn(η0)[0.006η4 0 - 0.0958η3 0 + 0.532η2 0 - 0.08η0] with an accuracy no less then 1%. The η0 is determined by η0 = T0/(T + δ), where T0 being degeneracy 4 A.I. ROMANENKO, O.B. ANIKEEVA, T.I. BURYAKOV... temperature of extrinsic carriers (EC) depends on its concentration n0 only. The value of δ can be estimated independently as δ = ℏ/πkBτ0 , where ℏis the Planck constant, τ0 is a relaxation time of the carrier nonthermally scattered by defects. Generally, the number of EC in QTDG is equal to that of scattering centers and δ depends only on T0, i.e. δ = T0/r, where r is determined by scattering efficiency. These parameters were chosen to give the best fit of the experimental data. According to theoretical calculations (Al’tshuler et al., 1983), the correction ∆χor to the orbital susceptibility χor in the Cooper channel dominates the quantum correction to the magnetic susceptibility χ(T, B) in magnetic fields smaller than Bc = (πkBT/gµB) (Bc = 9.8 T at 4.2 K). These corrections are determined by the value and the sign of the electron-electron interaction constant λc and are proportional to the diamagnetic susceptibility of electrons χor. In graphite and MWNTs the diamagnetic susceptibility is greater than in any other diamagnetic material (excluding the superconductors), and the", + "T at 4.2 K). These corrections are determined by the value and the sign of the electron-electron interaction constant λc and are proportional to the diamagnetic susceptibility of electrons χor. In graphite and MWNTs the diamagnetic susceptibility is greater than in any other diamagnetic material (excluding the superconductors), and the correction to χor should also be large. ∆χ(T)or = χ(T)exp or −χ(T)or was found by (Lee and Ramakrishnan, 1985; Al’tshuler et al., 1983): ∆χor(T) χor(T) = − 4 3(lel h )ln[ln(Tc T )] ln(kBTcτel ℏ ) , (d = 2), (2) ∆χor(T) χor(T) = − 2(π 6)ξ(1 2)(kBTτel ℏ )1/2 ln(Tc T ) , (d = 3), (3) where χ(T)exp or are the experimental data, χ(T)or is the result of an approx- imation of the experimental data in an interval of temperatures 50 - 400 K by the theoretically predicted dependence (1) for quasi-two-dimensional graphite (Kotosonov et al., 1997); value of ξ(1 2) ∼1, lel is the electron mean free path; τel represents the elastic relaxation time which is about 10−13 sec for MWNT (Baxendale et al., 1997); h is the thickness of graphene layers packet; d denotes the dimensionality of the system; Tc = θDexp(λ−1 c ), where θD is the Debye tem- perature, λc is the constant which describes the electron-electron interaction in the Cooper channel (λc > 0 in a case of electron repulsion). The dependence in Eq. (2) is determined by ln[ln(Tc T )] term because, at low temperatures, in the disordered systems, τel is temperature independent while all other terms are constants. The dependence in Eq. (3) is governed by T 1/2 term as Tc ≫T and, therefore, ln(Tc T ) can be considered as a constant relative to T 1/2. The temperature depen- dence of the magnetic susceptibility χ(T) is shown in figure 1 for arc-MWNTs before bromination, in figure 2 for arc-MWNTs after bromination, and in figure 3 for crystal graphite. Below 50 K the deviation of experimental data from the theoretical curve is observed (Romanenko et al., SSC, 2002; Romanenko et al., 2003). The additional contribution to χ(T) is presented in Fig. 1(b), 2(b), 3(b); and Fig. 1(c), 2(c), 3(c) as a function of ln[ln(Tc T )] and T 1/2 respectively. The ELECTRON-ELECTRON INTERACTION IN CARBON NANOSTRUCTURES 5 0 100 200 300 400 -20 -18 -16 -14 -12 -10 -8 -6 B = 0.01 T B = 0.5 T B = 5.5 T (10 -6 emu/g) T (K) (a) arc-MWNTs before bromination 0.50 0.75 1.00 1.25 -0.08 -0.06 -0.04 -0.02 0.00 arc-MWNTs before bromination B = 0.01 T B = 0.5 T B = 5.5 T (b) or or ln[ln(T)] 2 3 4 5 6 -0.08 -0.06 -0.04 -0.02 0.00 arc-MWNTs before bromination B = 0.01 T B = 0.5 T B = 5.5 T (c) or or T 1/2 (K 1/2 ) Figure 1. The temperature dependence of magnetic susceptibility χ(T) (a) and ∆χor(T)/χor(T) = [χ(T) −χor(T)]/χor(T) [(b) and (c)] for arc-produced MWNTs sample before bromination. The solid lines are fits: for (a) by Eq. (1) in interval 50 - 400 K with", + "T (c) or or T 1/2 (K 1/2 ) Figure 1. The temperature dependence of magnetic susceptibility χ(T) (a) and ∆χor(T)/χor(T) = [χ(T) −χor(T)]/χor(T) [(b) and (c)] for arc-produced MWNTs sample before bromination. The solid lines are fits: for (a) by Eq. (1) in interval 50 - 400 K with parameters; for curve (◦) , γ0 = 1.6 eV, T0 = 215 K, δ = 159 K; for (•) , γ0 = 1.6 eV, T0 = 215 K, δ = 159 K; for (⋆) , γ0 = 1.7 eV, T0 = 327 K, δ = 210 K; by Eq. (2) and Eq. (3) for (b) and (c) respectively in interval 4.5 - 45 K with parameters Tc = 10000 K , lel/a = 0.15. ∆χor(T)/χor(T) clearly shows the dependence given by Eq. (2) at low magnetic field and one given by Eq. (3) at high magnetic field, while at B = 0.5 T the temperature dependence of △χor(T)/χor(T) differs from those two limits. As seen from Fig. 1, at all magnetic fields applied to the arc-MWNTs before bromination, the absolute value of ∆χor(T)/χor(T) increases with decreasing temperature as has been predicted for IE in the systems characterized by electron-electron repulsion (Lee and Ramakrishnan, 1985; Al’tshuler et al., 1983). Hence, at B = 5.5 T a crossover from the two-dimensional IE correction to the three-dimensional one takes place. At lower magnetic field the interaction length LIE(T) is much shorter than the magnetic length LB, which in turn becomes dominant at high field. An estimation of the characteristic lengths gives respectively the value of LIE(4.2K) = 130 Å (taking into account that the diffusion constant D = 1 cm2/s (Baxendale et al., 1997)) and the value of LB = 100 Å at B = 5.5 T. A similar dependence of ∆χor(T)/χor(T) was observed for arc-MWNTs after bromination (Fig. 2). 6 A.I. ROMANENKO, O.B. ANIKEEVA, T.I. BURYAKOV... 0 100 200 300 400 -10 -9 -8 -7 -6 -5 -4 B = 0.01 T B = 0.5 T B = 5.5 T (10 -6 emu/g) T (K) (a) arc-MWNTs after bromination 0.50 0.75 1.00 1.25 -0.06 -0.04 -0.02 0.00 B = 0.01 T B = 0.5 T B = 5.5 T (b) or or ln[ln(T)] arc-MWNTs after bromination 2 3 4 5 6 -0.08 -0.06 -0.04 -0.02 0.00 B = 0.01 T B = 0.5 T B = 5.5 T (c) or or T 1/2 (K 1/2 ) arc-MWNTs after bromination Figure 2. The temperature dependence of magnetic susceptibility χ(T) (a) and ∆χor(T)/χor(T) = [χ(T) −χor(T)]/χor(T) [(b) and (c)] for arc-produced MWNTs sample after bromination. The solid lines are fits: for (a) by Eq. (1) in interval 50 - 400 K with parameters; for curve (◦) , γ0 = 1.4 eV, T0 = 340 K, δ = 252 K; for (•) , γ0 = 1.4 eV, T0 = 300 K, δ = 273 K; for (⋆) , γ0 = 1.5 eV, T0 = 435 K, δ = 325 K; by Eq. (2) and Eq. (3) for (b) and (c) respectively in interval 4.5 - 45 K", + "= 252 K; for (•) , γ0 = 1.4 eV, T0 = 300 K, δ = 273 K; for (⋆) , γ0 = 1.5 eV, T0 = 435 K, δ = 325 K; by Eq. (2) and Eq. (3) for (b) and (c) respectively in interval 4.5 - 45 K with parameters Tc = 10000 K , lel/a = 0.15. The dependence of ∆χor(T)/χor(T) we investigated for crystals of graphite (Fig. 3). However, only the three-dimensional dependence ∆χor(T)/χor(T) ≈T 1/2 was found for graphite. Approximation of the abnormal part of the magnetic susceptibility by theoret- ically predicted functions has revealed three features: I. A crossover from the two-dimensional quantum corrections to χ(T) in fields B = 0.01 T to the three-dimensional quantum corrections in field B = 5.5 T is observed as up to bromination of arc-MWNTs, so after bromination of its when the magnetic field increases. For graphite, in all fields, the three-dimensional corrections to χ(T) are observed. This is related to the fact that magnetic length LB = 77 Å in fields of B = 5.5 T becomes comparable to the thickness of the graphene layers hMWNT which form the MWNT. On the other hand, in fields of B = 0.01 T, LB = 1800 Å is much longer than hMWNT but it is shorter than the length of a tube lMWNT (lMWNT ≈1µ) and is comparable to the circumference of a nanotube. In graphite, the thickness of a package of the graphene layers is always ELECTRON-ELECTRON INTERACTION IN CARBON NANOSTRUCTURES 7 0 100 200 300 400 -30 -28 -26 -24 -22 -20 -18 -16 Crystal graphite (a) (10 -6 emu/g) T (K) 0.4 0.6 0.8 1.0 1.2 -0.14 -0.12 -0.10 -0.08 -0.06 Crystal graphite or / or ln(T/K) (b) 2 3 4 5 6 -0.14 -0.12 -0.10 -0.08 -0.06 Crystal graphite T 1/2 (K 1/2 ) or (T)/ or (0) (c) -2.0 -1.5 -1.0 2 3 4 5 6 or / or *[ln( D /T)+1/ c ] T 1/2 (K 1/2 ) (d) Crystal graphite Figure 3. Temperature dependences of a magnetic susceptibility χ(T) for graphite measured in a magnetic field = 0.01 T. Continuous lines show: the regular parts χ(T) (a); two-dimensional quantum corrections to χ(T) (b); three-dimensional quantum corrections to χ(T) (c), (d). The solid lines are fits for (d) by Eq. (3) in interval 4.5 - 45 K with parameters θD = 1000 K , λc = 0.1. macroscopic and it exceeds all other characteristic lengths. II. The bromination of arc-MWNTs has led to an increase of their conductance by one order of magnitude (from 500 Ω−1cm−1 in arc-MWNTs before bromi- nation up to 5000 Ω−1cm−1 after bromination). However, the relative correction to the magnetic susceptibility ∆χ(T)or/χ(T)or which determines the value of λc, remained constant. Thus, the bromination does not change the electron-electron interaction constant λc in the arc-produced multiwalled carbon nanotubes. III. The constant of electron-electron interaction λc for arc-MWNTs before and after bromination has the magnitude about 0.2 (Romanenko et al., SSC, 2002) which is greater than that of graphite λc ≈0.1 (Romanenko et", + "the bromination does not change the electron-electron interaction constant λc in the arc-produced multiwalled carbon nanotubes. III. The constant of electron-electron interaction λc for arc-MWNTs before and after bromination has the magnitude about 0.2 (Romanenko et al., SSC, 2002) which is greater than that of graphite λc ≈0.1 (Romanenko et al., 2003), i.e. the curvature of the graphene layers in MWNTs leads to the increase of λc. The temperature dependence of the electrical conductivity σ(T) of arc-MWNTs indicates the presence of quantum corrections also (Fig. 4(a) and Fig. 5(a)). At low temperatures, the temperature dependence of these quantum corrections is characteristic for the two-dimensional case (Fig. 4(b) and Fig. 5(b)): ∆σ(T) = ∆σWL(T) + ∆σIE(T) . (4) 8 A.I. ROMANENKO, O.B. ANIKEEVA, T.I. BURYAKOV... Here ∆σWL(T) ≈ln(Lϕ/lel) is the correction associated with the quantum interference of noninteracting electrons in two-dimensional systems (WL) (Lee and Ramakrishnan, 1985; Kawabata, 1980) while ∆σIE(T) ≈ln(LIE/lel) is the correction associated with the quantum interference of interacting electrons (IE) in such systems (Lee and Ramakrishnan, 1985; Al’tshuler et al., 1983). The contribu- tion of quantum corrections to the electrical conductivity should be accompanied by corrections to magnetoconductivity ∆σ(B) = 1/ρ(B) in low magnetic fields (Kawabata, 1980; Al’tshuler et al., 1981): ∆σ(B) = ∆σWL(B) + ∆σIE(B). (5) Here ∆σWL(B) is the quantum correction to magnetoconductance for nonin- teracting electrons; ∆σIE(B) - the quantum correction to the magnetoconductance for interacting electrons. Both corrections have the logarithmic asymptotic in high magnetic fields (∆σWL(B) ≈ln(Lϕ/LB); ∆σInt(B) ≈ln(LIE/LB) at Lϕ/lB; LIE/LB >> 1), and the quadratic asymptotic in low magnetic fields (∆σWL(B) ≈ B2; ∆σIE(B) ≈B2 when Lϕ/LB; LIE/LB << 1). The quantum corrections to magnetoconductance become essential in low magnetic fields when the magnetic length is LB < Lϕ. In this case the phase of an electron is lost at distances ≈LB, and quantum corrections to conductance are partially suppressed. This leads to positive magnetoconductance (negative magnetoresistance). It is difficult to divide the contribution of IE in the Cooper channel (which is determined by the amplitude and sign of λc), WL and IE in the diffusion channel. Field dependences of magnetoresistance (Fig. 4(c)) in all intervals of the measured fields (0 - 1 T) show negative magnetoresistance, related to WL. Similar depen- dences are observed in graphite (Fig. 4(d)). All three mechanisms give quantum corrections to temperature dependence of conductance σ(T). In order to observe the contribution of IE in the superconducting channel it is necessary to exclude the contribution of WL. We achieved this in catalytic carbon multiwalled nanotubes which were synthesized by a technology which prevents the formation of other phases of carbon. 3.2. CATALYTIC MULTIWALLED CARBON NANOTUBES There are always paramagnetic impurities present in c-MWNTs because of the ferromagnetic metals used as catalytic agents. It is not possible to use the magnetic susceptibility in order to obtain information on ∆χor(T) of c-MWNT, because the contribution of paramagnetic impurities dominates at low temperatures. We carried out an analysis of the conductivity data for which the contribution of paramagnetic impurities is negligible. In figure 5(c) the dependence of the relative magnetoconductivity", + "use the magnetic susceptibility in order to obtain information on ∆χor(T) of c-MWNT, because the contribution of paramagnetic impurities dominates at low temperatures. We carried out an analysis of the conductivity data for which the contribution of paramagnetic impurities is negligible. In figure 5(c) the dependence of the relative magnetoconductivity ρ(B)/ρ(0) on the magnetic field B for c-MWNTs is shown. The closed circles - c-MWNTs prepared by a usual catalytic method (Ku- dashov et al., 2002); the open circles - with use of the special procedure (Couteau ELECTRON-ELECTRON INTERACTION IN CARBON NANOSTRUCTURES 9 0 50 100 150 200 250 300 150 200 250 300 (1/ cm) T (K) (a) 1.5 2.0 2.5 3.0 3.5 4.0 -14 -12 -10 -8 -6 -4 -2 0 - extr (1/ cm) ln(T/K) (b) -1.0 -0.5 0.0 0.5 1.0 0.94 0.95 0.96 0.97 0.98 0.99 1.00 (B)/ (0) B (T) (c) -1.0 -0.5 0.0 0.5 1.0 0.90 0.92 0.94 0.96 0.98 1.00 (B)/ (0) B (T) (d) Figure 4. Data for arc-produced MWNTs (a, b, c) and for pyrolytic graphite (d). Temperature dependence of conductivity σ(T) (a), anomaly part of conductivity ∆σ(T) = σ(T) −σ(T)ext (b), and the relative magnetoconductivity σ(B)/σ(0) from magnetic field B measured at T = 4.2 K (c, d). Continuous lines show: σ(T)ext receiving by extrapolation of approximation curve from T ≥50 K to T ≤50 K (a); two-dimensional quantum corrections to σ(T) (b), asymptotic of quadratic approximation σ(B)/σ(0) ≈B2 at B ≤0.05 T to B up to 0.3 T (c, d). Dashed lines on (c) show the logarithmic asymptotic σ(B)/σ(0) ≈ln(T) from high field to low field. et al., 2003) which allows to prepare c-MWNTs practically without inclusions of the another form of carbon. Comparing these curves it is possible to see, that in the refined samples in the low magnetic fields (at B ≤0.2 T) the contribution to the negative magnetoconductivity, related to WL, we not observe. σ(B)/σ(0) for refined c-MWNTs is described by quadratic dependence σ(B)/σ(0) ≈B2 for all values of field B. σ(B)/σ(0) for c-MWNT prepared by a usual method are described by quadratic dependence only in low magnetic fields (B < 0.1 T). In figure 5(d) the dependence of the relative magnetoconductivity σ(B)/σ(0) on the magnetic field B is shown for soot. As can be seen from figure 5 the curves for soot and c-MWNTs prepared by a usual catalytic method are very similar at low fields. We suggest that this fact is connected to the presence of impurities of soot in c-MWNTs prepared by a usual catalytic method. Investigation of σ(T) of the c-MWNTs shows the presence of quantum corrections σ(T) ≈ln(T) and 10 A.I. ROMANENKO, O.B. ANIKEEVA, T.I. BURYAKOV... 0 50 100 150 200 250 300 15 20 25 (1/ cm) T (K) (a) 2 3 4 -6 -4 -2 0 - extr (1/ cm) ln(T/K) (b) -1.0 -0.5 0.0 0.5 1.0 0.996 0.997 0.998 0.999 1.000 (B)/ (0) B (T) -1.0 -0.5 0.0 0.5 1.0 0.94 0.95 0.96 0.97 0.98 0.99 1.00 (B)/ (0) B (T) (d) Figure 5. Data for: ◦- catalytic MWNTs without", + "4 -6 -4 -2 0 - extr (1/ cm) ln(T/K) (b) -1.0 -0.5 0.0 0.5 1.0 0.996 0.997 0.998 0.999 1.000 (B)/ (0) B (T) -1.0 -0.5 0.0 0.5 1.0 0.94 0.95 0.96 0.97 0.98 0.99 1.00 (B)/ (0) B (T) (d) Figure 5. Data for: ◦- catalytic MWNTs without impurities of another forms of carbon (a, b, c); • - MWNTs prepared by a usual catalytic method(c), and ∆- soot (d). Temperature dependence of conductivity σ(T) (a), anomaly part of conductivity ∆σ(T) = σ(T) −σ(T)ext (b), and the relative magnetoconductivity σ(B)/σ(0) from magnetic field B measured at T = 4.2 K (c, d). Continuous lines show: σ(T)ext receiving by extrapolation of approximation curve from T ≥50 K to T ≤50 K (a), two-dimensional quantum corrections to σ(T) (b), asymptotic of quadratic approximation σ(B)/σ(0) ≈B2 from B ≤0.05 T to B up to 0.3 T for (curve • on figure c) and soot (d). Dashed line: on (c) - show the quadratic approximation σ(B)/σ(0) ≈B2, on (d) - show the logarithmic asymptotic σ(B)/σ(0) ≈ln(T) from high field to low field. suggests the two-dimensional character of these corrections (Kawabata, 1980; Lee and Ramakrishnan, 1985; Al’tshuler et al., 1983). The magnitude of the magnetic field which suppresses the temperature correction (δσ(4.2 K)/σ(4.2 K) ≈2.7%) is estimated as B ≈8.5 T. This corresponds to a quite reasonable magnitude of magnetic length LB ≈60 Å. Negative magnetoconductivity for IE is also in agreement with the conclusion that λc > 0. However this result has been already obtained from the resistance data. 4. Conclusion Analysis of the anomalous part of the magnetic susceptibility χ at temperatures below 50 K has allowed us to estimate the electron-electron interaction constant, ELECTRON-ELECTRON INTERACTION IN CARBON NANOSTRUCTURES 11 λc, in the arc-produced MWNTs (λc ≈0.2) and in graphite (λc ≈0.1). We found that λc does not change in arc-produced MWNTs when the concentration of cur- rent carriers is modified by bromination. The analysis of the anomalous part of the conductance and the positive magnetoconductivity demonstrate a dominating of contribution of weak localization effects. In pure catalytic MWNTs we found positive magnetoconductivity related only to interaction effects which points to the positive sign of λc in these nanotubes. Thus, the analysis of temperature and field dependences of the magnetic susceptibility, conductivity and magnetocon- ductivity allows us to estimate the electron-electron interaction constant, λc, and to determine the effective dimensionality of the current carriers in inhomoge- neous systems. On the base of our results we can conclude that the curvature of graphine layers in carbon nanostructures is responsible for the change of constant electron-electron interaction λc. Acknowledgements The work was supported by Russian Foundation of Basic Research (Grants No: 05-03-32901, 06-02-16005); Russian Ministry of Education and Sciences (Grant PHΠ.2.1.1.1604); Joint Grant CRDF, and Russian Ministry of Education and Sci- ences (NO-008-X1). References Al’tshuler, B. L., Aronov, A. G., and Zyuzin A. Yu. (1983) Thermodynamic properties of disordered conductors, Sov. Phys. JETP 57, 889–895. Al’tshuler, B. L., Aronov, A.G., Larkin, A.I., and Khmel’nitski, D.E. (1981) Anomalous magne- toresistance in semiconductors, Sov. Phys. JETP 54,", + "Russian Ministry of Education and Sci- ences (NO-008-X1). References Al’tshuler, B. L., Aronov, A. G., and Zyuzin A. Yu. (1983) Thermodynamic properties of disordered conductors, Sov. Phys. JETP 57, 889–895. Al’tshuler, B. L., Aronov, A.G., Larkin, A.I., and Khmel’nitski, D.E. (1981) Anomalous magne- toresistance in semiconductors, Sov. Phys. JETP 54, 411–419. Baxendale, M., Mordkovich, V.Z., Yoshimura, S., and Chang, R.P.H. (1997) Magnetotransport in bundles of intercalated carbon nanotubes, Phys. Rev. B 56, 2161–2165. Couteau, E., Hernadi, K., Seo, J.W., Thien-Nga, L., Miko, C., Gaal, R., Forro, L. (2003) CVD synthesis of high-purity multiwalled carbon nanotubes using CaCO3 catalyst support for large- scale production, Chem. Phys. Lett. 378, 9-17. Gonzalez, J., Guinea, F., and Vozmediano, M. A. H. (2001) Electron-electron interactions in graphene sheets, Phys. Rev. B 63, 134421-1 –134421-8. Kawabata A. (1980) Theory of negative magnetoresistance in three-dimensional systems, Solid State Commun. 34, 431–432. Kociak, M., Kasumov, A.Yu., Guron, S., Reulet, B., Khodos, I. I., Gorbatov, Yu. B., Volkov, V. T., Vaccarini, L., and Bouchiat, H. (2001) Superconductivity in Ropes of Single-Walled Carbon Nanotubes, Phys. Rev. Lett. 86, 2416–2419. Kopelevich, Y., Esquinazi, P., Torres,J. H. S., and Moehlecke, S. (2000) Ferromagnetic- and Superconducting-Like Behavior of Graphite, Journal of Low Temperature Physics 119 5–6. Kotosonov, A. S., Kuvshinnikov, S. V. (1997) Diamagnetism of some quasi-two-dimensional graphites and multiwall carbon nanotubes, Phis. Lett. A 229 377–380. 97) Diamagnetism of some quasi-two-dimensional graphites and multiwall carbon nanotubes, Phys. Lett. A 229, 377–380 12 A.I. ROMANENKO, O.B. ANIKEEVA, T.I. BURYAKOV... Kudashov, A. G., Okotrub, A. V., Yudanov, N. F., Romanenko, A. I., Bulusheva, L. G., Abrosimov, O. G., Chuvilin, A. L., Pazhetov, E. M., Boronin, A. I. (2002) Gas-phase synthesis of nitrogen- containing carbon nanotubes and their electronic properties, Physics of Solid State 44 652–655. Kudashov, A. G., Abrosimov, O. G., Gorbachev, R. G., Okotrub, A. V., Yudanova L. I., Chuvilin, A. L., Romanenko, A. I. (2004) Comparison of structure and conductivity of multiwall car- bon nanotubes obtained over Ni and Ni/Fe catalysts, Fullerenes, Nanotubes, and Carbon Nanostructures 12, 93–97. Lee P A, Ramakrishnan T. V. (1985) Disordered electronic systems, Rev. Modern Phys. 57, 287– 337. Mott, N. F. (1979) Electron Processes in Noncrystalline Materrials, Oxford, Clarendon Press, 350 p. Okotrub, A.V., Bulusheva, L.G., Romanenko, A.I., Chuvilin, A.L., Rudina, N.A., Shubin, Y.V., Yudanov, N.F., Gusel’nikov, A.V. (2001) Anisotropic properties of carbonaceous material produced in arc discharge, Appl. Phys. A 71 481-486. Okotrub, A. V., Bulusheva, L. G., Romanenko, A. I., Kuznetsov, V.L., Butenko, Yu.V., Dong, C., Ni, Y., Heggic, M.I. (2001) Probing the electronic state of onion-like carbon. in Electronic Properties of Molecular Nanostructures (AIP Conference Proceedings, New York, 2001) 591 p.349–352. Romanenko, A. I., Anikeeva, O. B., Okotrub, A. V., Bulusheva, L. G., Yudanov, N. F., Dong, C., and Ni, Y. (2002) Transport and Magnetic Properties of Multiwall Carbon Nanotubes before and after Bromination, Physics of Solid State 44, 659–662. Romanenko, A. I., Okotrub, A. V., Anikeeva, O. B., Bulusheva, L. G., Yudanov, N. F., Dong, C., Ni, Y. (2002) Electron-electron interaction in multiwall carbon nanotubes, Solid State Commun. 121, 149–153. Romanenko, A. I., Anikeeva,", + "Properties of Multiwall Carbon Nanotubes before and after Bromination, Physics of Solid State 44, 659–662. Romanenko, A. I., Okotrub, A. V., Anikeeva, O. B., Bulusheva, L. G., Yudanov, N. F., Dong, C., Ni, Y. (2002) Electron-electron interaction in multiwall carbon nanotubes, Solid State Commun. 121, 149–153. Romanenko, A. I., Anikeeva, O. B., Okotrub, A. V., Kuznetsov, V.L., Butenko, Yu.V., Chuvilin, A.L., Dong, C., Ni, Y. (2002) Diamond nanocomposites and onion-like carbon in Nanophase and nanocomposite materials, vol. 703, Eds. S. Komarneni, J.-I. Matsushita, G.Q. Lu, J.C. Parker, R.A. Vaia, Material Research Society, (Pittsburgh 2002) p. 259–264 Temperature dependence of electroresistivity, negative and positive magnetoresistivity of graphite. Romanenko, A. I., Okotrub, A. V., Bulusheva, L. G., Anikeeva, O. B., Yudanov, N. F., Dong, C., Ni, Y. (2003) Impossibility of superconducting state in multiwall carbon nanotubes and single crystal graphite, Physica C 388-389, 622–623. Romanenko, Romanenko, Anikeeva, O. B., Zhmurikov, E. I., Gubin, K. V., Logachev, P.V., Dong, C., Ni, Y. (2004) Influence of the structural defects on the electrophysical and magnetic proper- ties of carbon nanostructures, in The Progresses In Function Materials (11th APAM Conference proceedings, Ningbo, P. R. China, 2004) 59–61. Takesue, I., Haruyama, J., Kobayashi, N., Chiashi, S., Maruyama, S., Sugai, T., Shinohara, H., (2006) Superconductivity in Entirely End-Bonded Multiwalled Carbon Nanotubes, Phys. Rev. Lett. 96, 057001-1–057001-4. Tang, Z.K., Zhang, L.Y., Wang, N., Zhang, X.X., Wang, J.N., Li, G.D., Li, Z.M., Wen, G.H., Chan, C.T., Sheng, P. (2003) Ultra-small single-walled carbon nanotubes and their superconductivity properties, Synthetic metals 133-134, 689–693", + "arXiv:physics/0602037v4 [physics.gen-ph] 30 Jul 2007 The rest masses of the electron and muon and of the stable mesons and baryons E.L. Koschmieder Center for Statistical Mechanics The University of Texas at Austin, Austin TX 78712, USA e-mail: koschmieder@mail.utexas.edu The rest masses of the electron, the muon and of the stable mesons and baryons can be explained, within 1% accuracy, with the standing wave model, which uses only photons, neutrinos, charge and the weak nuclear force. We do not need hypothetical particles for the explanation of the masses of the electron, muon, mesons and baryons. We have determined the rest masses of the electron-, muon- and tau neutrinos and found that the mass of the electron neutrino is equal to the fine structure constant times the mass of the muon neutrino. Key words: Neutrino masses, electron mass, muon mass, meson masses, baryon masses. PACS numbers: 14.40.-n; 14.60.-z; 14.60.Pq. Introduction The so-called “Standard Model” of the elementary particles has, until now, not come up with a precise theoretical determination of the masses of either the mesons and baryons or of the leptons, which means that neither the mass of the fundamental electron nor the mass of the fundamental proton have been explained. The quarks, which have been introduced by Gell-Mann [1] forty years ago, are said to explain the mesons and baryons. But the standard model does not explain neither the mass, nor the charge, nor the spin of the particles, the three fundamental time-independent properties of the particles. The fractional electric charges imposed on the quarks do not explain the charge e±, neither does the spin imposed on the quarks explain the spin. The measured values of the properties of the particles are given in the Review of Particle Physics [2]. There are many other attempts to explain the elementary particles or only one of the particles, too many to list them 1 here. For example El Naschie has, in the last years, proposed a topological theory for high energy particles and the spectrum of the quarks [3-6]. The need for the present investigation has been expressed by Feynman [7] as follows: “There remains one especially unsatisfactory feature: the observed masses of the particles, m. There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we do not understand them - what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem”. Today, twenty years later, we still stand in front of the same problem. 1 The spectrum of the masses of the particles As we have done before [8] we will focus attention on the so-called “stable” mesons and baryons whose masses are reproduced with other data in Tables 1 and 2. It is obvious that any attempt to explain the masses of the mesons and baryons should begin with the particles that are affected by the fewest parameters. These are certainly the particles without isospin (I = 0) and without spin (J = 0), but also with strangeness S", + "and 2. It is obvious that any attempt to explain the masses of the mesons and baryons should begin with the particles that are affected by the fewest parameters. These are certainly the particles without isospin (I = 0) and without spin (J = 0), but also with strangeness S = 0, and charm C = 0. Looking at the particles with I,J,S,C = 0 it is startling to find that their masses are quite close to integer multiples of the mass of the π0 meson. It is m(η) = (1.0140 ± 0.0003) · 4m(π0), and the mass of the resonance η′ is m(η′) = (1.0137 ± 0.00015) · 7m(π0). Three particles seem hardly to be sufficient to establish a rule. However, if we look a little further we find that m(Λ) = 1.0332 · 8m(π0) or m(Λ) = 1.0190 · 2m(η). We note that the Λ particle has spin 1/2, not spin 0 as the π0, η mesons. Nevertheless, the mass of Λ is close to 8m(π0). Furthermore we have m(Σ0) = 0.9817 · 9m(π0), m(Ξ0) = 0.9742 · 10m(π0), m(Ω−) = 1.0325 · 12m(π0) = 1.0183 · 3m(η), (Ω− is charged and has spin 3/2). Finally the masses of the charmed baryons are m(Λ+ c ) = 0.9958 · 17m(π0) = 1.024 · 2m(Λ), m(Σ0 c) = 1.0093 · 18m(π0), m(Ξ0 c) = 1.0167 · 18m(π0), and m(Ω0 c) = 1.0017 · 20m(π0). Now we have enough material to formulate the integer multiple rule of the particle masses, according to which the masses of the η, Λ, Σ0, Ξ0, Ω−, Λ+ c , Σ0 c, Ξ0 c, and Ω0 c particles are, in a first approximation, integer multiples of the mass of the π0 meson, although some of the particles have spin, and may also have charge as well as strangeness and charm. A consequence of the integer multiple rule must be that the ratio of the mass of any meson or baryon listed above divided by the mass of another meson or baryon listed above is equal 2 Table 1: The ratios m/m(π0) of the π0 and η mesons and of the baryons of the γ-branch. m/m(π0) multiples decays fraction spin mode (%) π0 1.0000 1.0000 · π0 γγ 98.798 0 (1.) e+e−γ 1.198 η 4.0559 1.0140 · 4π0 γγ 39.43 0 (2.) 3π0 32.51 π+π−π0 22.6 π+π−γ 4.68 Λ 8.26577 1.0332 · 8π0 pπ− 63.9 1 2 2∗(2.) 1.0190 · 2η nπ0 35.8 Σ0 8.8352 0.9817 · 9π0 Λγ 100 1 2 2∗(2.) + (1.) Ξ0 9.7417 0.9742 · 10π0 Λπ0 99.52 1 2 2∗(2.) + 2(1.) Ω− 12.390 1.0326 · 12π0 ΛK− 67.8 3 2 3∗(2.) 1.0183 · 3η Ξ0π− 23.6 Ξ−π0 8.6 Λ+ c 16.928 0.9958 · 17π0 many 1 2 2∗(2.) + (3.) 0.9630 · 17π± Σ0 c 18.167 1.0093 · 18π0 Λ+ c π− ≈100 1 2 Λ+ c + π− Ξ0 c 18.302 1.0167 · 18π0 eleven (seen) 1 2 2∗(3.) Ω0 c 20.033 1.0017 · 20π0 seven (seen) 1 2 2∗(3.) + 2(1.) 0.9857 · 5η 1The modes apply to neutral particles", + "17π± Σ0 c 18.167 1.0093 · 18π0 Λ+ c π− ≈100 1 2 Λ+ c + π− Ξ0 c 18.302 1.0167 · 18π0 eleven (seen) 1 2 2∗(3.) Ω0 c 20.033 1.0017 · 20π0 seven (seen) 1 2 2∗(3.) + 2(1.) 0.9857 · 5η 1The modes apply to neutral particles only. The ∗marks coupled modes. 3 to the ratio of two integer numbers. And indeed, for example m(η)/m(π0) is practically two times (exactly 0.9950 · 2) the ratio m(Λ)/m(η). There is also the ratio m(Ω−)/m(Λ) = 0.9993 · 3/2. We have furthermore e.g. the ratios m(Λ)/m(η) = 1.019 · 2, m(Ω−)/m(η) = 1.018 · 3, m(Λ+ c )/m(Λ) = 1.02399 · 2, m(Σ0 c)/m(Σ0) = 1.0281 · 2, m(Ω0 c)/m(Ξ0) = 1.0282 · 2, and m(Ω0 c)/m(η) = 0.9857 · 5. We will call, for reasons to be explained soon, the particles discussed above, which follow in a first approximation the integer multiple rule, the γ-branch of the particle spectrum. The mass ratios of these particles are in Table 1. The deviation of the mass ratios from exact integer multiples of m(π0) is at most 3.3%, the average of the factors before the integer multiples of m(π0) of the nine γ-branch particles in Table 1 is 1.0066 ± 0.0184. From a least square analysis follows that the masses of the ten particles on Table 1 lie on a straight line given by the formula m(N)/m(π0) = 1.0065 N −0.0043 N > 1, (1) where N is the integer number nearest to the actual ratio of the particle mass divided by m(π0). The correlation coefficient in Eq.(1) has the nearly perfect value R2 = 0.999. The integer multiple rule applies to more than just the stable mesons and baryons. The integer multiple rule applies also to the γ-branch baryon resonances which have spin J = 1/2 and the meson resonances with I,J ≤ 1, listed in [2] or in Table 3 of the appendix. The Ω−baryon will not be considered because it has spin 3/2 but would not change the following equa- tion significantly. If we consider all mesons and baryons of the γ-branch in Table 3, “stable” or unstable, then we obtain from a least square analysis the formula m(N)/m(π0) = 0.999 N + 0.0867 N > 1, (2) with the correlation coefficient 0.9999. The line through the points is shown in Fig. 1 which tells that 41 particles of the γ-branch of different spin and isospin, strangeness and charm; five I,J = 0,0 η mesons, fifteen J = 1/2 baryons, ten I = 0, J = 0,1 c¯c mesons, ten I = 0, J = 0,1 b¯b mesons and the π0 meson with I,J = 1,0, lie on a straight line with slope 0.999. In other words they approximate the integer multiple rule very well. Spin 1/2 and spin 1 does not seem to affect the integer multiple rule, i.e. the ratios of the particle masses, neither does strangeness S ̸= 0 and charm C ̸= 0. 4 Fig. 1: The mass of the mesons and baryons of the γ-branch, stable", + "multiple rule very well. Spin 1/2 and spin 1 does not seem to affect the integer multiple rule, i.e. the ratios of the particle masses, neither does strangeness S ̸= 0 and charm C ̸= 0. 4 Fig. 1: The mass of the mesons and baryons of the γ-branch, stable or unstable, with I ≤1, J ≤1 in units of m(π0) as a function of the integer N, demonstrating the integer multiple rule. 5 Searching for what else the π0, η, Λ, Σ0, Ξ0, Ω−particles have in common, we find that the principal decays (decays with a fraction > 1%) of these par- ticles, as listed in Table 1, involve primarily γ-rays, the characteristic case is π0 →γγ (98.8%). We will later on discuss a possible explanation for the 1.198% of the decays of π0 which do not follow the γγ route but decay via π0 →e+ + e−+ γ. After the γ-rays the next most frequent decay product of the heavier particles of the γ-branch are π0 mesons which again decay into γγ. To describe the decays in another way, the principal decays of the par- ticles listed above take place always without the emission of neutrinos ; see Table 1. There the decays and the fractions of the principal decay modes are given, taken from the Review of Particle Physics. We cannot consider decays with fractions < 1%. We will refer to the particles whose masses are approximately integer multiples of the mass of the π0 meson, and which decay without the emission of neutrinos, as the γ-branch of the particle spectrum. To summarize the facts concerning the γ-branch. Within 0.66% on the average the masses of the stable particles of the γ-branch in Table 1 are integer multiples (namely 4, 8, 9, 10, 12, and even 17, 18, 20) of the mass of the π0 meson. It is improbable that nine particles have masses so close to integer multiples of m(π0) if there is no correlation between them and the π0 meson. It has, on the other hand, been argued that the integer multiple rule is a numerical coincidence. But the probability that the mass ratios of nine particles of the γ-branch fall by coincidence on integer numbers between 1 and 20 instead on all possible numbers between 1 and 20 with two decimals after the period is smaller than 10−20, i.e. nonexistent. The integer multiple rule is not affected by more than 3% by the spin, the isospin, the strangeness, and by charm. The integer multiple rule seems even to apply to the Ω−and Λ+ c particles, although they are charged. In order for the integer multiple rule to be valid the deviation of the ratio m/m(π0) from an integer number must be smaller than 1/2N, where N is the integer number closest to the actual ratio m/m(π0). That means that the permissible deviation decreases rapidly with increased N. All particles of the γ-branch have deviations smaller than 1/2N. The remainder of the stable mesons and baryons are the π±, K±,0, p, n, D±,0, and D± s particles", + "is the integer number closest to the actual ratio m/m(π0). That means that the permissible deviation decreases rapidly with increased N. All particles of the γ-branch have deviations smaller than 1/2N. The remainder of the stable mesons and baryons are the π±, K±,0, p, n, D±,0, and D± s particles which make up the neutrino-branch (ν-branch) of the particle spectrum. The ratios of their masses are given in Table 2. These particles are in general charged, exempting the K0 and D0 mesons and the neutron n, in contrast to the particles of the γ-branch, which are in general neutral. It does not make a significant difference whether one 6 Table 2: The ν-branch of the particle spectrum m/m(π±) multiples decays fraction spin mode (%) π± 1.0000 1.0000 · π± µ+νµ 99.9877 0 (1.) K±,0 3.53713 0.8843 · 4π± µ+νµ 63.43 0 (2.) + π0 π±π0 21.13 (K±) π+π−π+ 5.58 (2.) + π∓ π0e+νe (K+ e3) 4.87 (K0,K0) π0µ+νµ (K+ µ3) 3.27 n 6.73186 0.8415 · 8π± p e−νe 100. 1 2 2∗(2.) 0.9516 · 2K± + 2π± 0.9439 · (K0 + K 0) D±,0 13.393 0.8370 · 16π± e+ anything 17.2 0 2(2∗(2.) 0.9466 · 4K± K−anything 24.2 + 2π±) 0.9954 · (p + ¯n) K 0 anything + K0 anything 59 η anything < 13 K+ anything 5.8 D± s 14.104 0.8296 · 17π± K−anything 13 0 body 0.9968 · 4K± K 0 anything centered + K0 anything 39 cubic K+ anything 20 e+ anything 8 2The particles with negative charges have conjugate charges of the listed decays. Only the decays of K± and D± are listed. The oscillation modes carry one electric charge. The ∗ marks coupled modes. 7 considers the mass of a particular charged or neutral particle. After the π mesons, the largest mass difference between charged and neutral particles is that of the K mesons (0.81%), and thereafter all mass differences between charged and neutral particles are < 0.5%. The integer multiple rule does not immediately apply to the masses of the ν-branch particles if m(π±) (or m(π0)) is used as reference, because m(K±) = 0.8843 · 4m(π±). 0.8843 · 4 = 3.537 is far from integer. Since the masses of the π0 meson and the π± mesons differ by only 3.4% it has been argued that the π± mesons are, but for the isospin, the same particles as the π0 meson, and that therefore the π± mesons cannot start another particle branch. However, this argument is not supported by the completely different decays of the π0 mesons and the π± mesons. The π0 meson decays almost exclusively into γγ (98.8%), whereas the π± mesons decay practically exclusively into µ mesons and neutrinos, as in π+ →µ+ + νµ (99.9877%). Furthermore, the lifetimes of the π0 and the π± mesons differ by nine orders of magnitude, being τ(π0) = 8.4 · 10−17 sec versus τ(π±) = 2.6 · 10−8 sec. If we make the π± mesons the reference particles of the ν-branch, then we must multiply the mass ratios m/m(π±) of the above listed particles with an", + "the π± mesons differ by nine orders of magnitude, being τ(π0) = 8.4 · 10−17 sec versus τ(π±) = 2.6 · 10−8 sec. If we make the π± mesons the reference particles of the ν-branch, then we must multiply the mass ratios m/m(π±) of the above listed particles with an average factor 0.848 ± 0.025, as follows from the mass ratios on Ta- ble 2. The integer multiple rule may, however, apply directly if one makes m(K±) the reference for masses larger than m(K±). The mass of the neutron is 0.9516 · 2m(K±), which is only a fair approximation to an integer multi- ple. There are, on the other hand, outright integer multiples in m(D±) = 0.9954 · (m(p) + m(¯n)), and in m(D± s ) = 0.9968 · 4m(K±). A least square analysis of the masses of the ν-branch in Table 2 yields the formula m(N)/0.853m(π±) = 1.000 N + 0.00575 N > 1, (3) with R2 = 0.998. This means that the particles of the ν-branch are integer multiples of m(π±) times the factor 0.853. One must, however, consider that the π± mesons are not necessarily the perfect reference for all ν-branch particles, because π± has I = 1, whereas for example K± has I = 1/2 and S = ±1 and the neutron has also I = 1/2. Actually the factor 0.853 in Eq.(3) is only an average. The mass ratios indicate that this factor decreases slowly with increased m(N). The existence of the factor and its decrease will be explained later. Contrary to the particles of the γ-branch, the ν-branch particles decay preferentially with the emission of neutrinos, the foremost example is π± → 8 µ± + νµ(¯νµ) with a fraction of 99.9877%. Neutrinos characterize the weak interaction. We will refer to the particles in Table 2 as the neutrino branch (ν-branch) of the particle spectrum. We emphasize that a weak decay of the particles of the ν-branch is by no means guaranteed. Although the neutron decays via n →p + e−+ ¯νe in 887 sec (100%), the proton is stable. There are, on the other hand, weak decays such as e.g. K+ →π+π−π+ (5.59%), but the subsequent decays of the π± mesons lead to neutrinos and e±. To summarize the facts concerning the ν-branch of the mesons and bary- ons. The masses of these particles seem to follow the integer multiple rule if one uses the π± mesons as reference, however the mass ratios share a common factor 0.85 ± 0.025. To summarize what we have learned about the integer multiple rule: In spite of differences in charge, spin, strangeness, and charm the masses of the “stable” mesons and baryons of the γ-branch are integer multiples of the mass of the π0 meson within at most 3.3% and on the average within 0.66%. Correspondingly, the masses of the “stable” particles of the ν-branch are, after multiplication with a factor 0.85 ± 0.025, integer multiples of the mass of the π± mesons. The integer multiple rule has been anticipated much earlier by Nambu [9], who wrote in 1952", + "and on the average within 0.66%. Correspondingly, the masses of the “stable” particles of the ν-branch are, after multiplication with a factor 0.85 ± 0.025, integer multiples of the mass of the π± mesons. The integer multiple rule has been anticipated much earlier by Nambu [9], who wrote in 1952 that “some regularity [in the masses of the particles] might be found if the masses were measured in a unit of the order of the π-meson mass”. A similar suggestion has been made by Fr¨ohlich [10]. The integer multiple rule suggests that the particles are the result of superpositions of modes and higher modes of a wave equation. 2 Standing waves in a cubic lattice We will now study, as we have done in [11], whether the so-called “stable” particles of the γ-branch cannot be described by the frequency spectrum of standing waves in a cubic lattice, which can accommodate automatically the Fourier frequency spectrum of an extreme short-time collision by which the particles are created. The investigation of the consequences of lattices for particle theory was initiated by Wilson [12] who studied a cubic fermion lattice. His study has developed over time into lattice QCD. It will be necessary for the following to outline the most elementary as- pects of the theory of lattice oscillations. The classic paper describing lattice oscillations is from Born and v. Karman [13], henceforth referred to as B&K. They looked at first at the oscillations of a one-dimensional chain of points 9 with mass m, separated by a constant distance a. This is the monatomic case, all lattice points have the same mass. B&K assume that the forces exerted on each point of the chain originate only from the two neighboring points. These forces are opposed to and proportional to the displacements, as with elastic springs (Hooke’s law). The equation of motion is in this case m¨un = α(un+1 −un) −α(un −un−1) . (4) The un are the displacements of the mass points from their equilibrium posi- tion which are apart by the distance a. The dots signify, as usual, differenti- ation with respect to time, α is a constant characterizing the force between the lattice points, and n is an integer number. For a →0 Eq.(4) becomes the wave equation c2∂2u/∂x2 = ∂2u/∂t2 (B&K). In order to solve Eq.(4) B&K set un = Aei(ω t + nφ) , (5) which is obviously a temporally and spatially periodic solution or describes standing waves. n is an integer, with n < N, where N is the number of points in the chain. φ = 0 is the monochromatic case. We also consider higher modes, by replacing nφ in Eq.(5) by ℓnφ, with integer ℓ> 1. The wavelengths are then shorter by 1/ℓ. At nφ = π/2 are nodes, where for all times t the displacements are zero, as with standing waves f(x,t) = A cos(ωt) cos(nφ) = A cos(ωt) cos(kx). If a displacement is repeated after n points we have na = λ, where λ is the wavelength, a the lattice constant, and it must be nφ", + "nodes, where for all times t the displacements are zero, as with standing waves f(x,t) = A cos(ωt) cos(nφ) = A cos(ωt) cos(kx). If a displacement is repeated after n points we have na = λ, where λ is the wavelength, a the lattice constant, and it must be nφ = 2π according to (5). It follows that λ = 2πa/φ . (6) Inserting (5) into (4) one obtains a continuous frequency spectrum of the standing waves as given by Eq.(5) of B&K ω = ± 2 q α/m sin(φ/2) . (7) B&K point out that there is not only a continuum of frequencies, but also a maximal frequency which is reached when φ = π, or at the minimum of the possible wavelengths λ = 2a. The boundary conditions are periodic, that means that un = un+N, where N is the number of points in the chain. Born referred to the periodic boundary condition as a “mathematical convenience”. 10 The number of normal modes must be equal to the number of particles in the lattice. Born’s model of the crystals has been verified in great detail by X-ray scattering and even in much more complicated cases by neutron scattering. The theory of lattice oscillations has been pursued in particular by Blackman [14], a summary of his and other studies is in [15]. Comprehensive reviews of the results of linear studies of lattice dynamics have been written by Born and Huang [16], by Maradudin et al. [17], and by Ghatak and Kothari [18]. 3 The masses of the γ-branch particles We will now assume, as seems to be quite natural, that the particles consist of the same particles into which they decay, directly or ultimately. We know this from atoms, which consist of nuclei and electrons, and from nuclei, which consist of protons and neutrons. Quarks have never been observed among the decay products of elementary particles. For the γ-branch particles our assumption means that they consist of photons. Photons and π0 mesons are the principal decay products of the γ-branch particles, the characteristic example is π0 →γγ (98.8%). Table 1 shows that there are decays of the γ-branch particles which lead to particles of the ν-branch, in particular to pairs of π+ and π−mesons. It appears that this has to do with pair pro- duction in the γ-branch particles. Pair production is evident in the decay π0 →e+ + e−+ γ (1.198%) or in the π0 meson’s third most frequent decay π0 →e+e−e+e−(3.14·10−3%). Pair production requires the presence of elec- tromagnetic waves of high energy. Anyway, the explanation of the γ-branch particles must begin with the explanation of the most simple example of its kind, the π0 meson, which by all means seems to consist of photons. The composition of the particles of the γ-branch suggested here offers a direct route from the formation of a γ-branch particle, through its lifetime, to its decay products. Particles that are made of photons are necessarily neutral, as the majority of the particles of the γ-branch are. We also base our assumption that the", + "particles of the γ-branch suggested here offers a direct route from the formation of a γ-branch particle, through its lifetime, to its decay products. Particles that are made of photons are necessarily neutral, as the majority of the particles of the γ-branch are. We also base our assumption that the particles of the γ-branch are made of photons on the circumstances of the formation of the γ-branch particles. The most simple and straightforward creation of a γ-branch particle are the reactions γ + p →π0 + p, or in the case that the spins of γ and p are parallel γ + p →π0 + p + γ′. A photon impinges on a proton and creates a π0 meson. The considerations which follow apply as well for other photoproductions 11 such as γ + p →η + p or γ + d →π0 + d and to the photoproduction of Λ in γ + p →Λ + K+, but also for the electroproductions e−+ p →π0 + e−+ p or e−+ d →π0 + e−+ d, see Rekalo et al. [19]. The most simple example of the creation of a γ-branch particle by a strong interaction is the reaction p + p →p + p + π0. The electromagnetic energy accumulated in a proton during its acceleration reappears as the π0 meson. In γ + p →π0 + p the pulse of the incoming electromagnetic wave is in 10−23 sec converted into a continuum of electromagnetic waves with frequen- cies ranging from 1023 sec−1 to ν →∞according to Fourier analysis. There must be a cutofffrequency, otherwise the energy in the sum of the frequencies would exceed the energy of the incoming electromagnetic wave. The wave packet so created decays, according to experience, after 8.4 · 10−17 sec into two electromagnetic waves or γ-rays. It seems to be very unlikely that Fourier analysis does not hold for the case of an electromagnetic wave impinging on a proton. The question then arises of what happens to the electromagnetic waves in the timespan of 10−16 seconds between the creation of the wave packet and its decay into two γ-rays ? We will show that the electromag- netic waves can continue to exist for the 10−16 seconds until the wave packet decays. If the wave packet created by the collision of a γ-ray with a proton consists of electromagnetic waves, then the waves cannot be progressive because the wave packet must have a rest mass. The rest mass is the mass of a particle whose center of mass does not move. However standing electromagnetic waves have a rest mass. Standing electromagnetic waves are equivalent to a lattice, because in standing waves the waves travel back and forth between the nodes, just as lattice points oscillate between the nodes of the lattice oscillations. The oscillations in the lattice take care of the continuum of frequencies of the Fourier spectrum of the collision. So we assume that the very many photons in the wave packet are held together in a cubic lattice. It is not unprecedented that photons", + "the nodes of the lattice oscillations. The oscillations in the lattice take care of the continuum of frequencies of the Fourier spectrum of the collision. So we assume that the very many photons in the wave packet are held together in a cubic lattice. It is not unprecedented that photons have been considered to be building blocks of the elementary particles. Schwinger [20] has once studied an exact one-dimensional quantum electrodynamical model in which the photon acquired a mass ∼e2. On the other hand, it has been suggested by Sidharth [21] that the π0 meson consists of an electron and a positron which circle their center of mass. We will now investigate the standing waves in a cubic photon lattice. We assume that the lattice is held together by a weak force acting from one lattice point to its nearest neighbors. We assume that the range of this force is 10−16 cm, because the range of the weak nuclear force is on the order of 12 10−16 cm, as stated e.g. on p.25 of Perkins [22]. For the sake of simplicity we set the sidelength of the lattice at 10−13 cm, the exact size of the nucleon is given in [23] and will be used later. With a = 10−16 cm there are then 109 lattice points. As we will see the ratios of the masses of the particles are independent of the sidelength of the lattice. Because it is the most simple case, we assume that a central force acts between the lattice points. We cannot consider spin, isospin, strangeness or charm of the particles. The frequency equation for the oscillations in an isotropic monatomic cubic lattice with central forces is, in the one-dimensional case, given by Eq.(7). The direction of the oscillation is determined by the direction of the incoming wave. According to Eq.(13) of B&K the force constant α is α = a (c11 −c12 −c44) , (8) where c11, c12 and c44 are the elastic constants in continuum mechanics which applies in the limit a →0. If we consider central forces then c12 = c44 which is the classical Cauchy relation. Isotropy requires that c44 = (c11 −c12)/2. The waves are longitudinal. Transverse waves in a cubic lattice with concentric forces are not possible according to [18]. All frequencies that solve Eq.(7) come with either a plus or a minus sign which is, as we will see, important. The reference frequency in Eq.(7) is ν0 = q α/4π2m , (9) or as we will see, using Eq.(11), ν0 = c⋆/2πa. The limitation of the group velocity has now to be considered. The group velocity is given by cg = dω dk = a r α m · df(φ) dφ . (10) The group velocity of the photons has to be equal to the velocity of light c⋆ throughout the entire frequency spectrum, because photons move with the velocity of light. In order to learn how this requirement affects the frequency distribution we have to know the value of q α/m in a photon lattice. But we do not", + "to be equal to the velocity of light c⋆ throughout the entire frequency spectrum, because photons move with the velocity of light. In order to learn how this requirement affects the frequency distribution we have to know the value of q α/m in a photon lattice. But we do not have information about what either α or m might be in this case. In the following we set a q α/m = c⋆, which means, since a = 10−16 cm, that q α/m = 3 · 1026 sec−1, or that the corresponding period is τ = 1/3 · 10−26 13 sec, which is the time it takes for a wave to travel with the velocity of light over one lattice distance. With c⋆= a q α/m (11) the equation for the group velocity is cg = c⋆· df/dφ . (12) For photons that means, since cg must then always be equal to the velocity of light c⋆, that df/dφ = 1. This requirement determines the form of the frequency distribution regardless of the order of the mode of oscillation or it means that instead of the sine function in Eq.(7) the frequency is given by ν = ± ν0( φ + φ0 ) . (13) For the time being we will disregard φ0 in Eq.(13). The frequencies of the corrected spectrum in Eq.(13) must increase from ν = 0 at the origin φ = 0 with slope 1 (in units of ν0) until the maximum is reached at φ = π. The energy contained in the oscillations must be proportional to the sum of all frequencies (Eq.14). The second mode of the lattice oscillations contains 4 times as much energy as the basic mode, because the frequencies are twice the frequencies of the basic mode, and there are twice as many oscillations. Adding, by superposition, to the second mode different numbers of basic modes or of second modes will give exact integer multiples of the energy of the basic mode. Now we understand the integer multiple rule of the particles of the γ-branch. There is, in the framework of this theory, on account of Eq.(13), no alternative but integer multiples of the basic mode for the energy contained in the frequencies of the different modes or for superpositions of different modes. In other words, the masses of the different particles are integer multiples of the mass of the π0 meson, if there is no spin, isospin, strangeness or charm. We remember that the measured masses in Table 1, which incorporate different spins, isospins, strangeness and charm, spell out the integer mul- tiple rule within on the average 0.65% accuracy. It is worth noting that there is no free parameter if one takes the ratio of the energies contained in the frequency distributions of the different modes, because the factor q α/m in Eq.(7) or ν0 in Eq.(13) cancels. This means, in particular, that the ratios of the frequency distributions, or the mass ratios, are independent of the 14 mass of the photons at the lattice points, as well as of the magnitude", + "of the different modes, because the factor q α/m in Eq.(7) or ν0 in Eq.(13) cancels. This means, in particular, that the ratios of the frequency distributions, or the mass ratios, are independent of the 14 mass of the photons at the lattice points, as well as of the magnitude of the force between the lattice points. It is obvious that the integer multiples of the frequencies are only a first approximation of the theory of lattice oscillations and of the mass ratios of the particles. The equation of motion in the lattice Eq.(4) does not apply in the eight corners of the cube, nor does it apply to the twelve edges nor, in particular, to the six sides of the cube. A cube with 109 lattice points is not correctly described by the periodic boundary condition we have used to de- rive Eq.(7), but is what is referred to as a microcrystal. A phenomenological theory of the frequency distributions in microcrystals, considering in particu- lar the surface energy, can be found in Chapter 6 of Ghatak and Kothari [18]. The surface energy may account for the small deviations of the mass ratios of the mesons and baryons from the integer multiple rule of the oscillations in a cube. However, it seems to be futile to pursue a more accurate deter- mination of the oscillation frequencies before we know what the interaction of the electron with mass is. The mass of the electron is 0.378% of the mass of the π0 meson and hence is a substantial part of the deviation of the mass ratios from the integer multiple rule. Let us summarize our findings concerning the γ-branch. The particles of the γ-branch consist of standing electromagnetic waves. The π0 meson is the basic mode. The η meson corresponds to the second mode, as is suggested by m(η) ≈4m(π0). The Λ particle corresponds to the superposition of two second modes, as is suggested by m(Λ) ≈2m(η). This superposition apparently results in the creation of spin 1/2. The two modes would then have to be coupled. The Σ0 and Ξ0 baryons are superpositions of one or two basic modes on the Λ particle. The Ω−particle corresponds to the superposition of three coupled second modes as is suggested by m(Ω−) ≈ 3m(η). This procedure apparently causes spin 3/2. The charmed Λ+ c particle seems to be the first particle incorporating a third mode. Σ0 c is apparently the superposition of a negatively charged basic mode on Λ+ c , as is suggested by the decay of Σ0 c. The easiest explanation of Ξ0 c is that it is the superposition of two coupled third modes. The superposition of two modes of the same type is, as in the case of Λ, accompanied by spin 1/2. The Ω0 c baryon is apparently the superposition of two basic modes on the Ξ0 c particle. All neutral particles of the γ-branch are thus accounted for. The explanation of the charged γ-branch particles Σ± and Ξ−has been discussed in [56]. The modes of the particles are listed in", + "1/2. The Ω0 c baryon is apparently the superposition of two basic modes on the Ξ0 c particle. All neutral particles of the γ-branch are thus accounted for. The explanation of the charged γ-branch particles Σ± and Ξ−has been discussed in [56]. The modes of the particles are listed in Table 1. We have also found the γ-branch antiparticles, which follow from the 15 negative frequencies which solve Eq.(7) or Eq.(13). Antiparticles have al- ways been associated with negative energies. Following Dirac’s argument for electrons and positrons, we associate the masses with the negative fre- quency distributions with antiparticles. We emphasize that the existence of antiparticles is an automatic consequence of our theory. All particles of the γ-branch are unstable with lifetimes on the order of 10−10 sec or shorter. Born [24] has shown that the oscillations in cubic lattices held together by central forces are unstable. It seems, however, to be possible that the particles can be unstable for reasons other than the instability of the lattice which apparently causes the most frequent decay of the π0 meson π0 →γγ (98.798%), or the most frequent decay of the η meson η →γγ (39.43%). Pair production seems to make it possible to understand the decay of the π0 meson π0 →e−+ e+ + γ (1.198%). Since in our model the π0 meson consists of a multitude of electromagnetic waves it seems that pair production takes place within the π0 meson, and even more so in the higher modes of the γ-branch where the electrons and positrons created by pair production tend to settle on mesons, as e.g. in η →π++π−+π0 (22.6%) or in the decay η →π+ + π−+ γ (4.68%), where the origin of the pair of charges is more apparent. Pair production is also evident in the decays η → e+e−γ (0.6%) or η →e+e−e+e−(6.9·10−3%). Finally we must explain the reason for which the photon lattice or the γ-branch particles are limited in size to a particular value of about 10−13 cm, as the experiments tell. Conventional lattice theory using the periodic boundary condition does not limit the size of a crystal, and in fact very large crystals exist. If, however, the lattice consists of standing electromagnetic waves the size of the lattice is limited by the radiation pressure. The lattice will necessarily break up at the latest when the outward directed radiation pressure is equal to the inward directed elastic force which holds the lattice together. For details we refer to [25]. 4 The rest mass of the π0 meson So far we have studied the ratios of the masses of the particles. We will now determine the mass of the π0 meson in order to validate that the mass ratios link with the actual masses of the particles. The energy of the π0 meson is E(m(π0)) = 134.9766 MeV = 2.16258 · 10−4 erg. 16 The sum of the energies E = hν of the frequencies of all standing one- dimensional waves in π0 seems to be given by the equation Eν = Nhν0 2π π Z −π", + "The energy of the π0 meson is E(m(π0)) = 134.9766 MeV = 2.16258 · 10−4 erg. 16 The sum of the energies E = hν of the frequencies of all standing one- dimensional waves in π0 seems to be given by the equation Eν = Nhν0 2π π Z −π f(φ)dφ . (14) This equation originates from B&K. N is the number of all lattice points. The total energy of the frequencies in a cubic lattice is equal to the num- ber N of the oscillations times the average of the energy of the individual frequencies. In order to arrive at an exact value of N in Eq.(14) we have to use the correct value of the radius of the proton, which is rp = (0.880 ± 0.015) · 10−13 cm according to [23] or rp = (0.883 ± 0.014) · 10−13 cm accord- ing to [26]. With a = 10−16 cm it follows that the number of all lattice points in the cubic lattice is N = 2.854 · 109 ∼= 1 418 3 . (15) When the number of the grid points of a cubic lattice is derived from the volume of a sphere one cannot arrive at an integer number to the third. The radius of the π± mesons has also been measured [27] and after further analysis [28] was found to be 0.83 · 10−13 cm, which means that within the uncertainty of the radii we have rp = rπ. And according to [29] the charge radius of Σ−is (0.78 ± 0.10) · 10−13 cm. If the oscillations are parallel to an axis, the limitation of the group velocity is taken into account, that means if Eq.(13) applies and the absolute values of the frequencies are taken, then the value of the integral in Eq.(14) is π2. With N = 2.854 · 109 and ν0 = c⋆/2πa follows from Eq.(14) that the sum of the energy of the frequencies of the basic mode corrected for the group velocity limitation is Ecorr = 1.418 · 109 erg. That means that the energy is 6.56 · 1012 times larger than E(m(π0)). This discrepancy is inevitable, because the basic frequency of the Fourier spectrum after a collision on the order of 10−23 sec duration is ν = 1023 sec−1, which means, when E = hν, that one basic frequency alone contains an energy of about 9 m(π0)c2 ⋆. To eliminate this discrepancy we use, instead of the simple form E = hν, the complete quantum mechanical energy of a linear oscillator as given by Planck E = hν ehν/kT −1 . (16) 17 This equation was already used by B&K for the determination of the specific heat of cubic crystals or solids. Equation (16) calls into question the value of the temperature T in the interior of a particle. We determine T empirically with the formula for the internal energy of solids u = RΘ eΘ/T −1 , (17) which is from Sommerfeld [30]. In this equation R = 2.854 · 109 k, where k is Boltzmann’s constant, and Θ is", + "temperature T in the interior of a particle. We determine T empirically with the formula for the internal energy of solids u = RΘ eΘ/T −1 , (17) which is from Sommerfeld [30]. In this equation R = 2.854 · 109 k, where k is Boltzmann’s constant, and Θ is the characteristic temperature introduced by Debye [31] for the explanation of the specific heat of solids. It is Θ = hνm/k, where νm is a maximal frequency. In the case of the oscillations making up the π0 meson the maximal frequency is νm = πν0, therefore νm = 1.5 · 1026 sec−1, and we find that Θ = 7.2 · 1015 K. In order to determine T we set the internal energy u equal to m(π0)c2 ⋆. It then follows from Eq.(17) that Θ/T = 30.20, or T = 2.38 · 1014 K. That means that Planck’s formula (16) introduces a factor 1/(eΘ/T −1 ) ∼= 1/e30.2 = 1/(1.305·1013) into Eq.(14). In other words, if we determine the temperature T of the particle through Eq.(17), and correct Eq.(14) accordingly then we arrive at a sum of the oscillation energies in the π0 meson which is N X 1 Eν = 1.0866 · 10−4 erg = 67.82 MeV . (18) That means that the sum of the energies of the one-dimensional oscillations consisting of N waves is 0.502 E(m(π0)). We have to double this amount because standing waves consist of two waves traveling in opposite direction with the same absolute value of the frequency. The sum of the energy of the oscillations in the π0 meson is therefore Eν(π0)(theor) = 2.1732 · 10−4 erg = 135.64 MeV = 1.005 E(m(π0))(exp) , (19) if the oscillations are parallel to the φ axis. The energy in the measured mass of the π0 meson and the energy in the sum of the oscillations agree fairly well, considering the uncertainties of the parameters involved. The theoret- ical mass of the η meson is then m(η) = 4 · m(π0)(theor) = 542.56 MeV = 0.991 m(η)(exp), and the theoretical mass of the Λ baryon, the superposition of two η mesons, is then m(Λ)(theor) = 8 · m(π0)(theor) = 1085.1 MeV = 0.9726 m(Λ)(exp). 18 To sum up: The π0 meson is formed when a γ-ray collides with a proton, γ + p →π0 + p. By the collision the incoming γ-ray is converted into a packet of standing electromagnetic waves, the π0 meson. After 10−16 seconds the wave packet decays into two electromagnetic waves, π0 →γγ. Electro- magnetic waves prevail throughout the entire process. The energy in the rest mass of the π0 meson and the other particles of the γ-branch is correctly given by the sum of the energy of standing electromagnetic waves in a cube, if the energy of the oscillations is determined by Planck’s formula for the energy of a linear oscillator. The π0 meson is like an adiabatic, cubic black body filled with standing electromagnetic waves. A black body of a given size and tempera- ture can certainly contain the energy in the rest", + "the energy of the oscillations is determined by Planck’s formula for the energy of a linear oscillator. The π0 meson is like an adiabatic, cubic black body filled with standing electromagnetic waves. A black body of a given size and tempera- ture can certainly contain the energy in the rest mass of the π0 meson, which is O(10−4) erg, if only the frequencies are sufficiently high. We know from Bose’s work [32] that Planck’s formula applies to a photon gas as well. For all γ-branch particles we have found a simple mode of standing electomagnetic waves. Since the equation determining the frequency of the standing waves is quadratic it follows automatically that for each positive frequency there is also a negative frequency of the same absolute value, that means that for each particle there exists also an antiparticle. For the explanation of the sta- ble mesons and baryons of the γ-branch we use only photons, nothing else. This is a rather conservative explanation of the π0 meson and the γ-branch particles. We do not use hypothetical particles. From the frequency distributions of the standing waves follow the ratios of the masses of the particles which obey the integer multiple rule. It is important to note that in this theory the ratios of the masses of the γ-branch particles to the mass of the π0 meson do not depend on the sidelength of the lattice, and the distance between the lattice points, neither do they depend on the strength of the force between the lattice points nor on the mass of the lattice points. The mass ratios are determined only by the spectra of the frequencies of the standing electromagnetic waves. 5 The neutrino branch particles The masses of the neutrino branch, the π±, K±,0, n, D±,0 and D± s particles, are integer multiples of the mass of the π± mesons times a factor 0.85 ± 0.02 as we stated before. We assume, as appears to be quite natural, that the π± mesons and the other particles of the ν-branch consist of the same particles into which they decay, that means in the case of the π± mesons of muon neutrinos, 19 anti-muon neutrinos, electron neutrinos, anti-electron neutrinos and of an electron or positron, as exemplified by the decay sequence π± →µ± + νµ(¯νµ), µ± →e± + ¯νµ(νµ) + νe(¯νe). The absence of an electron neutrino νe in the decay branches of π−or of an anti-electron neutrino ¯νe in the decay branches of π+ can be explained with the composition of the electron or positron, which will be discussed in Section 9. The existence of neutrinos and antineutrinos is unquestionable. Since the particles of the ν-branch decay through weak decays, we assume, as appears likewise to be natural, that the weak nuclear force holds the particles of the ν-branch together. This assumption has far reaching consequences, it is not only fundamental for the explanation of the π± mesons, but leads also to the explanation of the µ± mesons and ultimately to the explanation of the mass of the electron. The existence of the", + "holds the particles of the ν-branch together. This assumption has far reaching consequences, it is not only fundamental for the explanation of the π± mesons, but leads also to the explanation of the µ± mesons and ultimately to the explanation of the mass of the electron. The existence of the weak nuclear force is unquestionable. Since the range of the weak interaction, which is about 10−16 cm [22], is only about a thousandth of the diameter of the particles, which is about 10−13 cm, the weak force can hold particles together only if the particles have a lattice structure, just as macroscopic crystals are held together by microscopic forces between atoms. In the absence of a force which originates in the center of the particle and affects all neutrinos of the particle the configuration of the particle is not spherical but cubic, reflecting the very short range of the weak nuclear force. We will show that the energy in the rest mass of the ν-branch particles is the energy in the oscillations of a cubic lattice consisting of electron and muon neutrinos and their antiparticles, plus the energy in the rest masses of the neutrinos. First it will be necessary to outline the basic aspects of diatomic lattice oscillations. In diatomic lattices the lattice points have alternately the masses m and M, as with the masses of the electron neutrinos m(νe) and muon neutrinos m(νµ). The classic example of a diatomic lattice is the salt crystal with the masses of the Na and Cl atoms in the lattice points. The theory of diatomic harmonic lattice oscillations was started by Born and v. Karman [13]. They first discussed a diatomic chain. The equation of motions in the chain are according to Eq.(22) of B&K m¨u2n = α(u2n+1 + u2n−1 −2u2n) , (20) M¨u2n+1 = α(u2n+2 + u2n −2u2n+1) , (21) where the un are the displacements, n an integer number and α a constant 20 characterizing the force between the particles. Eqs.(20,21) are solved with u2n = Aei(ω t + 2nφ), (22) u2n+1 = Bei(ω t + (2n+1)φ) , (23) where A and B are constants and φ is given by φ = 2πa/λ as in Eq.(6). a is the lattice constant as before and λ the wavelength, λ = na. The solutions of Eqs.(22,23) are obviously periodic in time and space and describe again standing waves. Using (22,23) to solve (20,21) leads to a secular equation from which according to Eq.(24) of B&K the frequencies of the oscillations of the chain follow from 4π2ν2 ± = α/Mm · ((M + m) ± q (M −m)2 + 4mMcos2φ ) . (24) Longitudinal and transverse waves are distinguished by the minus or plus sign in front of the square root in (24). 6 The masses of the ν-branch particles The characteristic case of the neutrino branch particles are the π± mesons which can be created in the process γ + p →π−+π+ + p. A photon impinges on a proton and is converted in 10−23 sec into a pair of particles of opposite charge.", + "masses of the ν-branch particles The characteristic case of the neutrino branch particles are the π± mesons which can be created in the process γ + p →π−+π+ + p. A photon impinges on a proton and is converted in 10−23 sec into a pair of particles of opposite charge. A simple example of the creation of a ν-branch particle by strong interaction is the case p + p →p + p + π−+ π+. Fourier analysis dictates that a continuum of frequencies must be in the collision products. The waves must be standing waves in order to be part of the rest mass of a particle. The π± mesons decay via π± →µ± + νµ(¯νµ) (99.98770%) followed by e.g. µ+ →e+ + ¯νµ + νe (≈100%). Only µ mesons, which decay into charge and neutrinos, and neutrinos result from the decay of the π± mesons. If the particles consist of the particles into which they decay, then the π± mesons are made of neutrinos, antineutrinos and e±. Since neutrinos interact through the weak force which has a range of 10−16 cm according to p.25 of [22], and since the size of the π± mesons [28] is on the order of 10−13 cm, the ν-branch particles must have a lattice with N neutrinos, N being the same as in Eq.(15). It is not known with certainty that neutrinos actually have a rest mass as was originally suggested by Bethe [33] and Bahcall [34] and what the values of m(νe) and m(νµ) are. However, the results of the Super-Kamiokande [35] and 21 the Sudbury [36] experiments indicate that the neutrinos have rest masses. The neutrino lattice must be diatomic, meaning that the lattice points have alternately larger (m(νµ)) and smaller (m(νe)) masses. We will retain the traditional term diatomic. The term neutrino lattice will refer to a lattice consisting of neutrinos and antineutrinos. The lattice we consider is shown in Fig. 2. Since the neutrinos have spin 1/2 this is a four-Fermion lattice which is required for the explanation of the weak decays. The first investigation of cubic Fermion lattices in context with the elementary particles was made by Wilson [12]. A neutrino lattice is electrically neutral. Since we do not know the interaction of the electron with a neutrino lattice we cannot consider lattices with a charge. Fig. 2: A cell in the neutrino lattice. Bold lines mark the forces between neutrinos and antineutrinos. Thin lines mark the forces between either neutrinos only, or antineutrinos only. A neutrino lattice takes care of the continuum of frequencies which must, according to Fourier analysis, be present after the high energy collision which created the particle. We will, for the sake of simplicity, first set the sidelength of the lattice at 10−13 cm that means approximately equal to the size of the nucleon. The lattice then contains about 109 lattice points. The sidelength of the lattice does not enter Eq.(24) for the frequencies of diatomic oscillations. The calculation of the ratios of the masses is consequently independent of the size of the lattice, as was", + "approximately equal to the size of the nucleon. The lattice then contains about 109 lattice points. The sidelength of the lattice does not enter Eq.(24) for the frequencies of diatomic oscillations. The calculation of the ratios of the masses is consequently independent of the size of the lattice, as was the case with the γ-branch. The size of the lattice can be explained with the pressure which the lattice oscillations exert on a crossection of the lattice. The pressure cannot exceed Young’s modulus of the lattice. We require that the lattice is isotropic. 22 0 0.5 1 1.5 2 nu_/nu0 0 45 90 135 180 phi Fig. 3: The frequency distribution ν−/ν0 of the basic diatomic mode according to Eq.(24) with M/m = 100. The dashed line shows the distribution of the frequencies corrected for the group velocity limitation. From the frequency distribution of the axial diatomic oscillations (Eq.24), shown in Fig. 3, follows the group velocity dω/dk = 2πa dν/dφ at each point φ. With ν = ν0f(φ) and ν0 = q α/4π2M = c⋆/2πa as in Eq.(9) we find cg = dω/dk = a q α/M · df(φ)/dφ . (25) In order to determine the value of dω/dk we have to know the value of q α/M. From Eq.(8) for α follows that α = a c44 in the isotropic case with central forces. The group velocity is therefore cg = q a3c44/M · df/dφ . (26) In Eq.(25) we now set a q α/M = c⋆, as in Eq.(11), where c⋆is the velocity of light. It follows that 23 cg = c⋆· df/dφ , (27) as it was with the γ-branch, only that now on account of the rest masses of the neutrinos the group velocity must be smaller than c⋆, so the value of df/dφ is limited to < 1, but cg ∼= c⋆, which is a necessity because the neutrinos in the lattice soon approach the velocity of light as we will see. Equation (27) applies regardless whether we consider ν+ or ν−in Eq.(24). That means that there are no separate transverse oscillations with their theoretically higher frequencies. The rest mass M of the heavy neutrino can be determined with lattice theory from Eq.(26) as we have shown in [11]. This involves the inaccurately known compression modulus of the proton. We will, therefore, rather deter- mine the rest mass of the muon neutrino with Eq.(29), which leads to m(νµ) ≈50 milli-eV/c2 ⋆. It can be verified easily that m(νµ) = 50 milli-eV/c2 ⋆makes sense. The energy of the rest mass of the π± mesons is 139 MeV, and we have N/4 = 0.7135·109 muon neutrinos and the same number of anti-muon neutrinos with an energy of about 50 milli-eV. It follows that the energy in the rest masses of all muon and anti-muon neutrinos is 71.35 MeV, that is 51% of the energy of the rest mass of the π± mesons, m(π±)c2 ⋆= 139.57 MeV. A very small part of m(π±)c2 ⋆goes, as we will see, into the electron neutrino masses, the rest of the energy in", + "masses of all muon and anti-muon neutrinos is 71.35 MeV, that is 51% of the energy of the rest mass of the π± mesons, m(π±)c2 ⋆= 139.57 MeV. A very small part of m(π±)c2 ⋆goes, as we will see, into the electron neutrino masses, the rest of the energy in π± is in the lattice oscillations. The energy in the rest mass of the π± mesons is the sum of the oscillation energies plus the sum of the energy in the rest masses of the neutrinos. The π± mesons are like cubic black bodies filled with oscillating neutrinos. For the sum of the energies of the frequencies we use Eq.(14) modified with Eq.(16), with the same N and ν0 we used for the γ-branch. For the integral in Eq.(14) of the axial diatomic frequencies corrected for the group velocity limitation we find the value π2/2 as can be easily derived from the plot of the corrected frequencies in Fig. 3. The value of the integral in Eq.(14) for the axial diatomic frequencies ν = ν0φ is 1/2 of the value π2 of the same integral in the case of axial monatomic frequencies, because in the latter case the increase of the corrected frequencies continues to φ = π, whereas in the diatomic case the increase of the corrected frequencies ends at π/2, see Fig. 3. We consider cg to be so close to c⋆that it does not change the value of the integral in Eq.(14) significantly. It can be calculated that the time average of the velocity of the electron neutrinos in the π± mesons is ¯v = 0.99994c⋆if m(νe) = 0.365 milli-eV/c2 ⋆as will be shown in Eq.(37). Consequently we find 24 that the sum of the energies of the corrected diatomic neutrino frequencies is 0.5433·10−4 erg = 33.91 MeV. We double this amount because we deal with standing waves or the superposition of two waves of the same energy and find with Eq.(18) that the energy of the neutrino oscillations in π± is Eν(π±) ∼= 1/2 · Eν(π0) = 67.82 MeV = 0.486 m(π±)c2 ⋆, (28) or that ≈1/2 of the energy of π± is in the oscillation energy Eν(π±). In order to determine the sum of the rest masses of the neutrinos we make use of Eν(π±) and obtain an approximate value of the rest mass of the muon neutrino from m(π±)c2 ⋆−Eν(π±) = X [m(νµ) + m(¯νµ) + m(νe) + m(¯νe)]c2 ⋆= 71.75 MeV , (29) that means that ≈1/2 of the energy of π± is in the neutrino rest masses. Since nothing else but the electric charge contributes to the rest mass of π± it appears that in a good approximation the oscillation energy in π± is equal to the energy in the sum of the neutrino rest masses in π±, i.e. Eν(π±) ∼= Σ m(neutrinos)c2 = N/2 · (m(νµ) + m(νe))c2 ∼= 1/2 · m(π±)c 2.(29a) If m(νe) ≪m(νµ) and m(νµ) = m(¯νµ), as we will justify later, we arrive with N/2 = 1.427·109 at m(νµ) ≈50 milli-eV/c2 ⋆. The sum of the energy of the", + "rest masses in π±, i.e. Eν(π±) ∼= Σ m(neutrinos)c2 = N/2 · (m(νµ) + m(νe))c2 ∼= 1/2 · m(π±)c 2.(29a) If m(νe) ≪m(νµ) and m(νµ) = m(¯νµ), as we will justify later, we arrive with N/2 = 1.427·109 at m(νµ) ≈50 milli-eV/c2 ⋆. The sum of the energy of the rest masses of all neutrinos Eq.(29) plus the oscillation energy Eq.(28) gives the theoretical rest mass of the π± mesons which is, since we used m(π±) in the determination of the neutrino rest masses with Eq.(29), equal to the experimental rest mass of 139.57 MeV/c2 ⋆. A cubic lattice and conservation of neutrino numbers during the reaction γ + p →π+ + π−+ p necessitates that the π+ and π−lattices contain just as many electron neutrinos as anti-electron neutrinos. If the lattice is cubic it must have a center neutrino (Fig. 4). Conservation of neutrino numbers requires furthermore that the center neutrino of π+ is matched by an antineutrino in π−. In the decay sequence of (say) the π−meson π−→µ−+ ¯νµ and µ−→e−+ νµ + ¯νe an electron neutrino νe does not appear. But since (N - 1)/4 electron neutrinos νe must be in the π−lattice it follows that (N - 1)/4 electron neutrinos must go with the electron emitted in the µ−decay. 25 Fig. 4: The center of a NaCl lattice. (After Born and Huang). We must now be more specific about N, which is an odd number, because a cubic lattice (Fig. 4) has a center particle, just as the NaCl lattice. In the π± mesons there are then (N - 1)/4 muon neutrinos νµ and the same number of anti-muon neutrinos ¯νµ, as well as (N - 1)/4 electron neutrinos νe and the same number of anti-electron neutrinos ¯νe, plus a center neutrino or antineutrino. We replace N - 1 by N′. Since N′ differs from N by only one in 109 we have N′ ∼= N. Although the numerical difference between N and N′ is negligible we cannot consider any integer number N because that would mean that there could be fractions of a neutrino. N′ is an even number, because each cell (Fig. 2) of the lattice (Fig. 4) consists of four pairs of neutrinos. The antiparticle of the π+ meson is the particle in which all frequencies of the neutrino lattice oscillations have been replaced by frequencies with the opposite sign, all neutrinos replaced by their antiparticles and the positive charge replaced by the negative charge. If, as we will show, the antineutrinos have the same rest mass as the neutrinos it follows that the antiparticle of the π+ meson has the same mass as π+ but opposite charge, i.e. is the π−meson. As we will see, the explanation of the mass of the π± mesons opens the door to the explanation of the mass of the muon and of the electron. Now we turn to the K mesons whose mass is m(K±) = 0.8843 · 4m(π±). The primary decay of the K± mesons K± →µ± +νµ(¯νµ) (63.5%) leads to the same end products as the", + "opens the door to the explanation of the mass of the muon and of the electron. Now we turn to the K mesons whose mass is m(K±) = 0.8843 · 4m(π±). The primary decay of the K± mesons K± →µ± +νµ(¯νµ) (63.5%) leads to the same end products as the π± meson decay π± →µ± +νµ(¯νµ) (99.98%). From this and the decay of the µ mesons we learn that the K mesons must, at least 26 partially, be made of the same four neutrino types as in the π± mesons namely of muon neutrinos, anti-muon neutrinos, electron neutrinos and anti-electron neutrinos and their oscillation energies. However the K± mesons cannot be solely the second mode of the lattice oscillations of the π± mesons, because the second mode of the π± mesons has an energy of E((2.)π±) = 4Eν(π±) + N/2 · (m(νµ) + m(νe)) c2 ⋆ ∼= 2m(π±)c2 ⋆+ 1/2 · m(π±)c2 ⋆= 348.92 MeV , (30) with 2Eν(π±) ∼= m(π±)c2 ⋆and N/2 · (m(νµ) + m(νe)) ∼= m(π±)/2 from Eqs. (29,29a). The 348.9 MeV characterize the second or (2.) mode of the π± mesons, which fails m(K±)c2 ⋆= 493.7 MeV by a wide margin. The concept that the K± mesons are alone a higher mode of the π± mesons also contradicts our point that the particles consist of the particles into which they decay. The decays K± →π± + π0 (21.13%), as well as K+ →π0 + e+ + νe (4.87%), called K+ e3, and K+ →π0 + µ+ + νµ (3.27%), called K+ µ3, make up 29.27% of the K± meson decays. A π0 meson figures in each of these decays. If we add the energy in the rest mass of a π0 meson m(π0)c2 ⋆= 134.97 MeV to the 348.9 MeV in the second mode of the π± mesons then we arrive at an energy of 483.9 MeV, which is 98.0% of m(K±)c2 ⋆. Therefore we conclude that the K± mesons consist of the second mode of the π± mesons plus a π0 meson or are the state (2.)π± + π0. Then it is natural that π0 mesons from the π0 component in the K± mesons are among the decay products of the K± mesons. The average factor 0.85 ± 0.025 in the ratios of the masses of the particles of the ν-branch to the mass of the π± mesons is a consequence of the neutrino rest masses. They make it impossible that the ratios of the particle masses are outright integer multiples because the particles consist of the energy in the neutrino oscillations plus the neutrino rest masses which are independent of the order of the lattice oscillations. Since the contribution in percent of the neutrino rest masses to the ν-branch particle masses decreases with increased particle mass the factor in front of the mass ratios of the ν-branch particles must decrease with increased particle mass. The K0,K0 mesons have a rest mass m(K0,K0) = 1.00809 m(K±), or it is m(K0,K0) = 0.99984 (m(K±) + αf · 4m(π±)). We obtain the K0 meson if we superpose onto the", + "the factor in front of the mass ratios of the ν-branch particles must decrease with increased particle mass. The K0,K0 mesons have a rest mass m(K0,K0) = 1.00809 m(K±), or it is m(K0,K0) = 0.99984 (m(K±) + αf · 4m(π±)). We obtain the K0 meson if we superpose onto the second mode of the π± mesons instead of a π0 meson a basic mode of the π± mesons with a charge opposite to the charge of the sec- ond mode of the π± meson. The K0 and K0 mesons, or the state (2.)π± + π∓, 27 is made of neutrinos and antineutrinos only, without a photon component, because the second mode of π± as well as the basic mode π∓consist of neu- trinos and antineutrinos only. The K0 meson has a measured mean square charge radius ⟨r2⟩= - 0.077 ± 0.010 fm2 according to [2], which can only be if there are two charges of opposite sign within K0, as this model implies. Since the mass of a π± meson is by 4.59 MeV/c2 ⋆larger than the mass of a π0 meson the mass of K0 should be larger than m(K±), and indeed m(K0) −m(K±) = 3.972 MeV/c2 ⋆according to [2]. Similar differences occur with m(D±) −m(D0) and m(Ξ0 c) −m(Ξ+ c ). The decay K0 S →π+ + π−(68.6%) creates directly the π+ and π−mesons which are part of the (2.)π± + π∓structure of K0 we have suggested. The decay K0 S →π0 + π0 (31.4%) apparently originates from the 2γ branch of electron positron annihilation. Both decays account for 100% of the decays of K0 S. The decay K0 L →3π0 (21.1%) apparently comes from the 3γ branch of electron positron annihilation. The two decays of K0 L called K0 µ3 into π± µ∓νµ (27.18%) and K0 e3 into π± e∓νe (38.79%) which together make up 65.95% of the K0 L decays apparently originate from the decay of the second mode of the π± mesons in the K0 structure, either into µ∓+ νµ or into e∓+ νe. The same types of decay, apparently tied to the (2.)π± mode, accompany also the K± decays K± →π±π0 (20.92%) in which, however, a π0 meson replaces the π± mesons in the K0 L decay products. Our rule that the particles consist of the particles into which they decay also holds for the K0 and K0 mesons. The explanation of the K0,K0 mesons with the state (2.)π± + π∓confirms that the state (2.)π± + π0 was the correct choice for the explanation of the K± mesons. The state (2.)π± + π∓is also crucial for the explanation of the absence of spin of the K0,K0 mesons, as we will see later. The neutron whose mass is m(n) = 0.95156 · 2m(K±) is either the super- position of a K+ and a K−meson or of a K0 meson and a K0 meson. As has been shown in [56], the spin rules out a neutron consisting of a K+ and a K−meson. On the other hand, the neutron can be the superposition of a K0 and a", + "position of a K+ and a K−meson or of a K0 meson and a K0 meson. As has been shown in [56], the spin rules out a neutron consisting of a K+ and a K−meson. On the other hand, the neutron can be the superposition of a K0 and a K0 meson which guarantees that the neutron consists of neutrinos without a photon component. In this case the neutron lattice contains at each lattice point a νµ, ¯νµ, νe, ¯νe neutrino quadrupole and there is a single quadrupole of positive and negative electric charges because in each K0 and K0 meson are neutrino pairs at the lattice points and each K0 and K0 meson carries a pair of opposite elementary electric charges. There must be opposite charges in the neutron because it has a mean square charge radius ⟨r2⟩= - 0.1161 fm2 [2]. The lattice oscillations in the neutron must be a coupled pair 28 in order for the neutron to have spin 1/2, just as the Λ baryon with spin 1/2 is a superposition of two η mesons. With m(K0)(theor) = m(K±) + 4 MeV/c2 = 487.9 MeV/c2 ⋆from above it follows that m(n)(theor) ≈2m(K0)(theor) ≈ 975.8 MeV/c2 ⋆= 1.04 m(n)(exp). The proton, whose mass is m(p) = 0.99862 m(n), does not decay and does not tell which particles it is made of. However, we learn about the structure of the proton through the decay of the neutron n →p + e−+ ¯νe (100%). An electron and one single anti-electron neutrino is emitted when the neutron decays and 1.29333 MeV are released. But there is no place for a permanent vacancy of a single missing neutrino and for a small amount of permanently missing oscillation energy in a nuclear lattice. As it appears all anti-electron neutrinos are removed from the structure of the neutron in the neutron decay and converted into the kinetic energy of the decay products. This type of process will be explained in the following Section. On the other hand, it is certain that the proton consists of a neutrino lattice because the neutron has a neutrino lattice. The proton carries a net positive elementary electric charge because the neutron carries an e+e−e+e−quadrupole, of which one e− is lost in the β-decay. The concept that the proton carries just one elementary electric charge has been abandoned a long time ago when it was said that the proton consists of three quarks carrying fractional electric charges. Each elementary charge in the proton has a magnetic moment, all of them point in the same direction because the spin of the one e−must be opposite to the spin of the two e+. Each magnetic moment of the elementary charges has a g-factor ∼= 2. All three electric charges in the proton must then have a g-factor ≈6, whereas the measured g-factor of the magnetic moment of the proton is g(p) = 5.585 = 0.93 · 6. The D± mesons with m(D±) = 0.9954·(m(p) + m(¯n)) are the superposi- tion of a proton and an antineutron of opposite spin or of their", + "must then have a g-factor ≈6, whereas the measured g-factor of the magnetic moment of the proton is g(p) = 5.585 = 0.93 · 6. The D± mesons with m(D±) = 0.9954·(m(p) + m(¯n)) are the superposi- tion of a proton and an antineutron of opposite spin or of their antiparticles, whereas the superposition of a proton and a neutron with the same spin creates the deuteron with spin 1 and a mass m(d) = 0.9988 (m(p) + m(n)). In this case the proton and neutron interact with the strong force, never- theless the deuteron consists of a neutrino lattice with standing waves. The D± s mesons seem to be made of a body centered cubic lattice (Fig. 5) as dis- cussed in [38]. Summing up: The particles of the ν-branch consist of oscillating neutrinos and one or more positive and/or negative elementary electric charges. The characteristic feature of the ν-branch particles is the cubic lattice consisting of νµ, ¯νµ, νe, ¯νe neutrinos. The rest masses of the ν-branch particles is the sum 29 Fig. 5: A body-centered cell. (After Born and Huang). In the center of a D± s cell is a τ neutrino in the corners are νµ,¯νµ,νe,¯νe neutrinos. of the rest masses of the neutrinos and antineutrinos in the lattice plus the mass in the energy of the lattice oscillations. The existence of the neutrino lattice is a necessity if one wants to explain the spin, or the absence of spin, of the ν-branch particles. We do not use hypothetical particles for the explanation of the ν-branch particles, just as we did not use hypothetical particles for the explanation of the γ-branch particles. 7 The rest mass of the muon Surprisingly one can also explain the mass of the µ± mesons with the standing wave model. The µ mesons belong to the lepton family which is distinguished from the mesons and baryons by the absence of strong interaction with the mesons and baryons. The charged leptons make up 1/2 of the number of stable elementary particles. The standard model of the particles does not deal with the lepton masses. Barut [39] has given a simple and quite accurate empirical formula relating the masses of the electron, µ meson and τ meson, which formula has been extended by Gsponer and Hurni [40] to the quark masses. The origin of most of the µ± mesons is the decay of the π± mesons, π± →µ± +νµ(¯νµ), or the decay of the K± mesons, K± →µ± +νµ(¯νµ). The mass of the muons is m(µ±) = 105.658 369 ± 9·10−6 MeV/c2 ⋆, according to the Review of Particle Physics [2]. The mass of the muons is usually compared to the mass of the electron and is very often said to be m(µ±) = m(e±) · (1 + 3/2αf) = 206.554 m(e±), (αf being the fine structure constant), whereas 30 the experimental value is 206.768 m(e±). This formula for m(µ±) was given by Barut [41] following an earlier suggestion by Nambu [9] that m(µ±) ≈ m(e±) · 3/2αf. The muons are “stable”, their lifetime τ(µ±)", + "· (1 + 3/2αf) = 206.554 m(e±), (αf being the fine structure constant), whereas 30 the experimental value is 206.768 m(e±). This formula for m(µ±) was given by Barut [41] following an earlier suggestion by Nambu [9] that m(µ±) ≈ m(e±) · 3/2αf. The muons are “stable”, their lifetime τ(µ±) = 2.19703 · 10−6 ± 4 · 10−11 sec is about a hundred times longer than the lifetime of the π± mesons, that means longer than the lifetime of any other elementary particle, but for the electrons, protons and neutrons. Comparing the mass of the µ± mesons to the mass of the π± mesons from which the µ± mesons emerge we find, with m(π±) = 139.570 18 MeV/c2 ⋆, that m(µ±)/m(π±) = 0.757027 = 1.00937 · 3/4 ∼= 3/4 + αf = 0.757297 = 1.00036 · m(µ±)/m(π±)(exp). The term + αf in the preceding equation will be explained later, Eq.(71). The mass of the µ± mesons is in a good approximation 3/4 of the mass of the π± mesons. We have also m(π±) −m(µ±) = 33.9118 MeV/c2 ⋆= 0.24297m(π±) or approximately 1/4 · m(π±). The mass of the electron is approximately 1/207 of the mass of the muon, the contribution of m(e±) to m(µ±) will therefore be neglected in the following. We assume, as we have done be- fore and as appears to be natural, that the particles, including the muons, consist of the particles into which they decay. The µ± mesons decay via µ± → e± + ¯νµ(νµ) + νe(¯νe) (≈100%). The muons are apparently composed of an elementary electric charge and some of the neutrinos, antineutrinos and their oscillations which are, according to the standing wave model, present in the cubic neutrino lattice of the π± mesons from which the µ± mesons come. The µ± mesons with a mass m(µ±) ∼= 3/4 · m(π±) seem to be related to the π± mesons rather than to the electron with which the µ± mesons have been compared traditionally, although m(µ±) is separated from m(e±) by a factor ∼= 207. From Eq.(29) followed that the rest mass of a muon neutrino should be about 50 milli-eV/c2 ⋆. Provided that the mass of an electron neutrino m(νe) is small as compared to m(νµ), as will be shown by Eq.(42), we find, with N = 2.854·109, that: (a) The difference of the rest masses of the µ± and π± mesons is nearly equal to the sum of the rest masses of all muon, respectively anti-muon, neutrinos in the π± mesons. m(π±) −m(µ±) = 33.912 MeV/c2 ⋆ versus N′/4 · m(νµ) ≈35.68 MeV/c2 ⋆. 31 (b) The energy in the oscillations of all νµ, ¯νµ, νe, ¯νe neutrinos in the π± mesons is nearly the same as the energy in the oscillations of all ¯νµ, νe, ¯νe, respectively νµ, ¯νe, νe, neutrinos in the µ± mesons. The oscillation energy is the rest mass of a particle minus the sum of the rest masses of all neutrinos in the particle as in Eq.(29). With m(νµ) = m(¯νµ) and m(νe) = m(¯νe) from Eqs.(38,41) we have Eν(π±) =", + "¯νe, respectively νµ, ¯νe, νe, neutrinos in the µ± mesons. The oscillation energy is the rest mass of a particle minus the sum of the rest masses of all neutrinos in the particle as in Eq.(29). With m(νµ) = m(¯νµ) and m(νe) = m(¯νe) from Eqs.(38,41) we have Eν(π±) = m(π±)c2 ⋆−N′/2 · [m(νµ) + m(νe)]c2 ⋆= 68.22 MeV (31) versus Eν(µ±) = m(µ±)c2 ⋆−N′/4 · m(νµ)c2 ⋆−N′/2 · m(νe)c2 ⋆= 69.98 MeV . (32) Equation (32) means that either all N′/4 muon neutrinos or all N′/4 anti- muon neutrinos have been removed from the π± lattice in its decay. If, e.g., any νµ neutrinos were to remain in the µ+ meson after the decay of the π+ meson they ought to appear in the decay of µ+, but they do not. We attribute the 1.768 MeV difference between the left and right side of (a) to the second order effects which cause the deviations of the masses of the particles from the integer multiple rule. There is also the difference that the left side of (a) deals with two charged particles, whereas the right side deals with neutral particles. (b) seems to say that the oscillation energy of all neutrinos in the π± lattice is conserved in the π± decay, which seems to be necessary because the oscillation frequencies in π± and µ± must follow Eq.(13) as dictated by the group velocity limitation. If indeed Eν(π±) = Eν(µ±) (33) then it follows from the difference of Eqs.(31) and (32) that m(π±) −m(µ±) = N′/4 · m(νµ) = N′/4 · m(¯νµ) . (34) The N′/4 electron neutrinos, respectively anti-electron neutrinos, which come, as we will see in Section 9, into the µ± mesons with the elementary electric charge, are the recipients of 1/4 of the oscillation energy Eν(π±) of the π± mesons which becomes available when the muon neutrinos or anti-muon neutrinos leave the π± lattice in the π± decay. The neutrinos coming with e± make it possible that Eν(π±) = Eν(µ±). After the π± decay the muon neutrinos in µ± retain their original oscillation energy Eν(π±)/4, the electron neutrinos and anti-electron neutrinos in µ± retain their original oscillation 32 energies Eν(π±)/4 as well. The remaining oscillation energy Eν(π±)/4 of the π± lattice so far not accounted for is picked up by the electron neutrinos, respectively anti-electron neutrinos, brought into µ± by the elementary elec- tric charge. Without a recipient for this oscillation energy Eν(π±)/4 a stable new particle can apparently not be formed in the π± decay, that means there is no µ0 meson. We should note that in the π± decays only one single muon neutrino is emitted, not N′/4 of them, but that in the π± decay 33.912 MeV are released. Since according to (b) the oscillation energy of the neutrinos in the π± mesons is conserved in their decay the 33.912 MeV released in the π± decay can come from no other source then from the rest masses of all muon or all anti-muon neutrinos in the π± mesons. The average kinetic energy of the neutrinos in the", + "the neutrinos in the π± mesons is conserved in their decay the 33.912 MeV released in the π± decay can come from no other source then from the rest masses of all muon or all anti-muon neutrinos in the π± mesons. The average kinetic energy of the neutrinos in the π± lattice is about 50 milli-eV, so it is not possible for a single neutrino in π± to possess an energy of 33.9 MeV. The 33.9 MeV can come only from the sum of the muon neutrino rest masses. However, what happens then to the neutrino numbers ? Either conservation of neutrino numbers is violated or the decay energy comes from equal numbers of muon and anti- muon neutrinos. Equal numbers N′/8 muon and anti-muon neutrinos would then be in the µ± mesons. This would not make a difference in either the oscillation energy or in the sum of the rest masses of the neutrinos or in the spin of the µ± mesons. The sum of the spin vectors of the N′/4 muon or anti-muon neutrinos converted into kinetic energy is zero, as will become clear in Section 12. Inserting m(π±) −m(µ±) = N′/4 · m(νµ) from Eq.(34) into Eq.(31) we arrive at an equation for the theoretical value of the mass of the µ± mesons. It is m(µ±)c2 ⋆= 1/2 · [ Eν(π±) + m(π±)c2 ⋆+ N′ m(νe)c2 ⋆/2 ] = 103.95 MeV , (35) which is 0.9838 m(µ±)c2 ⋆(exp) and expresses m(µ±) through the well-known mass of π±, the calculated oscillation energy of π±, and a small contribution (0.4%) of the electron neutrino and anti-electron neutrino masses. Eq.(35) shows that our explanation of the mass of the µ± mesons comes close to the experimental value m(µ±) = 105.658 MeV/c2 ⋆. With Eν(π±) = Eν(µ±) and with m(π±) from Eq.(31) we find a different form of Eq.(35) which is, in the case of the µ+ meson, m(µ+) = Eν(µ±)/c2 ⋆+ N′m(¯νµ)/4 + N′m(νe)/2 . (36) 33 As Eq.(36) tells, the rest mass of the muons is the sum of the rest masses of the muon neutrinos, respectively anti-muon neutrinos, and of the masses of the electron- and anti-electron neutrinos which are in the muon lattice, plus the oscillation energy of these neutrinos, neglecting the mass of e±. The ratio m(µ±)/m(π±) is 3/4, as it must be, if we divide Eq.(36) by m(π±) which follows from Eq.(29) and if we neglect the small masses of the electron and anti-electron neutrinos. The µ mesons cannot be point particles because they have a neutrino lattice. The commonly held belief that the µ mesons are point particles is based on the results of scattering experiments. But at a true point the density of a “point particle” would be infinite, which poses a problem. Since, on the other hand, neutrinos do not interact, in a very good approximation, with charge or mass it will not be possible to determine the size of the µ meson lattice through conventional scattering experiments. The µ mesons appear to be point particles because only the electric charge of the muons", + "the other hand, neutrinos do not interact, in a very good approximation, with charge or mass it will not be possible to determine the size of the µ meson lattice through conventional scattering experiments. The µ mesons appear to be point particles because only the electric charge of the muons participates in the scattering process and the elementary electric charge scatters like a point particle. 99.5% of the muon mass consists of non-interacting neutrinos. There is therefore no measurable difference in e−-p and µ−-p scattering. Finally we must address the question for what reason do the muons or leptons not interact strongly with the mesons and baryons ? We have shown in [8] that a strong force emanates from the sides of a cubic lattice caused by the unsaturated weak forces of about 106 lattice points at the surface of the lattice of the mesons and baryons. This follows from the study of Born and Stern [42] which dealt with the forces between two parts of a cubic lattice cleaved in vacuum. The strong force between two particles is an automatic consequence of the weak internal force which holds the particles together. If the muons have a lattice consisting of one type of muon neutrinos, say, ¯νµ and of νe and ¯νe neutrinos their lattice surface is not the same as the surface of the cubic νµ, ¯νµ, νe, ¯νe lattice of the mesons and baryons in the standing wave model. Therefore the muon lattice does not bond with the cubic lattice of the mesons and baryons. To summarize what we have learned about the µ± mesons. Eq.(36) says that the energy in m(µ±)c2 ⋆is the sum of the oscillation energies plus the sum of the energy of the rest masses of the neutrinos and antineutrinos in m(µ±), neglecting the energy in e±. The three neutrino types in the µ± mesons (Fig. 6) are the remains of the cubic neutrino lattice in the π± mesons. Since all νµ respectively all ¯νµ neutrinos have been removed from the π± lattice in the π± decay the rest mass of the µ± mesons must be ∼= 3/4·m(π±), in agree- 34 ment with the experimental results. The µ± mesons are not point particles. Fig. 6: A section through the central part of the neutrino lattice of the µ+ meson without its charge. The mass of the τ ± mesons follows from the most frequent decay of the D± s mesons, D± s →τ ± + ντ(¯ντ) (6.4%). It can be shown readily that the oscillation energies of the lattices in D± s and in τ ± are the same. From that follows that the energy in the rest mass of the τ ± mesons is the sum of the oscillation energy in the τ meson lattice plus the sum of the energy of the rest masses of all neutrinos and antineutrinos in the τ meson lattice, just as with the µ± mesons. We will skip the details. The tau mesons are not point particles either. 8 The neutrino masses Now we come to the neutrino masses.", + "plus the sum of the energy of the rest masses of all neutrinos and antineutrinos in the τ meson lattice, just as with the µ± mesons. We will skip the details. The tau mesons are not point particles either. 8 The neutrino masses Now we come to the neutrino masses. There is no certain knowledge what the neutrino masses are. Numerous values for m(νe) and m(νµ) have been proposed and upper limits for them have been established experimentally which have, with time, decreased steadily. The Review of Particle Physics 35 [2] gives for the mass of the electron neutrino the value < 2 eV/c2. Nei- ther the Superkamiokande [35] nor the Sudbury [36] experiments determine a neutrino mass, however, both experiments make it very likely that the neu- trinos have rest masses. We will now determine the neutrino masses from the composition of the π± mesons and from the β-decay of the neutron. If the same principle that applies to the decay of the π± mesons, namely that in the decay the oscillation energy of the decaying particle is conserved and that an entire neutrino type supplies the energy released in the decay, also applies to the decay of the neutron n →p + e−+ ¯νe, then the mass of the anti-electron neutrino can be determined from the known difference ∆= m(n) −m(p) = 1.293 332 MeV/c2 ⋆[2]. Nearly one half of ∆comes from the energy lost by the emission of the electron, whose mass is 0.510 9989 MeV/c2 ⋆. N anti-electron neutrinos are in the neutrino quadrupoles in the neutron, one- fourth of them is carried away by the emitted electron. We have seen in the paragraph below Eq.(29) that the decay sequence of the π± mesons requires that the electron carries with it N′/4 electron neutrinos, if the π± mesons consist of a lattice with a center neutrino or antineutrino and equal numbers of νe, ¯νe, νµ, ¯νµ neutrinos as required by conservation of neutrino number dur- ing the creation of π±. The electron can carry N′/4 anti-electron neutrinos as well as N′/4 electron neutrinos. Since, as we will see shortly, m(νe) = m(¯νe) this does not make a difference energetically but is relevant for the orientation of the spin vector of the emitted electron. After the neutron has lost N′/4 anti-electron neutrinos to the electron emitted in the β-decay the other 3/4·N′ anti-electron neutrinos in the neutron provide the energy ∆− m(e−)c2 ⋆= 0.782 333 MeV released in the decay of the neutron. After division by 3/4·N′ the rest mass of the anti-electron neutrino is m(¯νe) = 0.365 milli eV/c2 ⋆. (37) Since theoretically the antineutron decays as ¯n →¯p + e+ + νe it follows from the same considerations as with the decay of the neutron that m(νe) = m(¯νe) . (38) We note that N′/4 · m(νe) = N′/4 · m(¯νe) = 0.51 m(e±) . (39) This equation is, as we will see, fundamental for the explanation of the mass of the electron. 36 Inserting Eq.(37) into Eq.(29) we find that m(νµ) = 49.91 milli eV/c2 ⋆.", + "= m(¯νe) . (38) We note that N′/4 · m(νe) = N′/4 · m(¯νe) = 0.51 m(e±) . (39) This equation is, as we will see, fundamental for the explanation of the mass of the electron. 36 Inserting Eq.(37) into Eq.(29) we find that m(νµ) = 49.91 milli eV/c2 ⋆. (40) Since the same considerations apply for either the π+ or the π−meson it follows that m(νµ) = m(¯νµ) . (41) Experimental values for the rest masses of the different neutrino types are not available. However, it appears that for the νµ ↔νe oscillation the value for ∆m2 = m2 2 −m2 1 = 3.2×10−3 eV2 given on p.1565 of [35] can be used to determine m2 = m(νµ) if m1 = m(νe) is much smaller than m2. We have then m(νµ) ≈56.56 milli-eV/c2 ⋆, which is compatible with the value of m(νµ) given in Eq.(40). From Eqs.(37,40) follows that m(νe) = 1/136.74 · m(νµ) ∼= αfm(νµ) . (42) 1/136.74 is 1.0021 times the fine structure constant αf = e2/¯hc = 1/137.036. It does not seem likely that Eq.(42) is just a coincidence. The probability for this being a coincidence is zero considering the infinite pool of numbers on which the ratio m(νe)/m(νµ) could settle. The mass of the τ neutrino ντ can be determined from the decay D± s →τ ± + ντ (¯ντ), and the subsequent decay τ ± →π± + ¯ντ(ντ), which are stated in [2]. The appearance of ντ in the decay of D± s and the presence of νµ, ¯νµ, νe, ¯νe neutrinos in the π± decay product of the τ ± mesons means that ντ, ¯ντ, νµ, ¯νµ, νe, ¯νe neutrinos must be in the D± s lattice. The additional ντ and ¯ντ neutrinos can be accomodated in the D± s lattice by a body-centered cubic lattice, in which there is in the center of each cubic cell one particle different from the particles in the eight cell corners (Fig. 5). In a body-centered cubic lattice are N′/4 cell centers, if N′ is the number of lattice points without the cell centers. If the particles in the cell centers are tau neutrinos then N′/8 ντ and N′/8 ¯ντ neutrinos must be present, because of conservation of neutrino numbers. From m(D± s ) = 1968.5 MeV/c2 ⋆and m(τ ±) = 1777 MeV/c2 ⋆follows that m(D± s ) −m(τ ±) = 191.5 MeV/c2 ⋆= N′/8 · m(ντ) . (43) The rest mass of the τ neutrinos is therefore m(ντ) = m(¯ντ) = 0.537 eV/c2 ⋆. (44) 37 From the neutrino masses given by Eq.(44) and Eq.(40) follows that m(ντ) = 10.76 m(νµ) = 1.048 (αw/αf)m(νµ) , (45) where αw is the weak coupling constant αw = g2 w/4π¯hc = 1.02·10−5(mW/mp)2 [22] and αf is the fine structure constant. We keep in mind that g2 w in αw is not nearly as accurately known as e2 in αf. With Eq.(42) we find that m(ντ) = 1.048 · αw/α2 f · m(νe) = 1474 m(νe) . (46) To summarize what we have learned about the masses of the leptons: We", + "We keep in mind that g2 w in αw is not nearly as accurately known as e2 in αf. With Eq.(42) we find that m(ντ) = 1.048 · αw/α2 f · m(νe) = 1474 m(νe) . (46) To summarize what we have learned about the masses of the leptons: We have found an explanation for the mass of the µ± mesons and τ ± mesons. We have also determined the rest masses of the e, µ, τ neutrinos and antineutri- nos. In other words, we have found the masses of all leptons, exempting the electron, which will be dealt with in the next Section. 9 The electron The electron differs from the other particles we have considered in so far as it appears that its electric charge cannot be separated from its mass, whereas in the other charged particles the mass of the electric charge is, in a first approximation, unimportant for the mass of the particles. Even in the muons the mass of the electron contributes only five thousandth of the muon mass. On the other hand, the electron is fundamental for the stability of the charged particles whose lifetime is sometimes orders of magnitude larger than the lifetime of their neutral counterparts. For example the lifetime of the π± mesons is eight orders of magnitude larger than the lifetime of π0, the lifetime of the proton is infinite, whereas the neutron decays in about 900 seconds and, as a startling example, the lifetime of Σ± is O(10−10) seconds, whereas the lifetime of Σ0 is O(10−20) seconds. There is something particular to the interaction of electric charge with the particle masses. After J.J. Thomson [43] discovered the small corpuscle which soon be- came known as the electron an enormous amount of theoretical work has been done to explain the existence of the electron. Some of the most distin- guished physicists have participated in this effort. Lorentz [44], Poincar´e [45], Ehrenfest [46], Einstein [47], Pauli [48], and others showed that it is fairly certain that the electron cannot be explained as a purely electromagnetic particle. In particular it was not clear how the elementary electric charge 38 could be held together in its small volume because the internal parts of the charge repel each other. Poincar´e [49] did not leave it at showing that such an electron could not be stable, but suggested a solution for the problem by introducing what has become known as the Poincar´e stresses whose origin however remained unexplained. These studies were concerned with the static properties of the electron, its mass m(e±) and its electric charge e. In order to explain the electron with its existing mass and charge it appears to be necessary to add to Maxwell’s equations a non-electromagnetic mass and a non-electromagnetic force which could hold the electric charge together. We shall see what this mass and force is. The discovery of the spin of the electron by Uhlenbeck and Goudsmit [50] increased the difficulties of the problem in so far as it now had also to be explained how the angular momentum ¯h/2", + "could hold the electric charge together. We shall see what this mass and force is. The discovery of the spin of the electron by Uhlenbeck and Goudsmit [50] increased the difficulties of the problem in so far as it now had also to be explained how the angular momentum ¯h/2 and the magnetic moment µe come about. The spin of a point-like electron seemed to be explained by Dirac’s [51] equation, however it turned out later [52] that Dirac type equa- tions can be constructed for any value of the spin. Afterwards Schr¨odinger [53] tried to explain the spin and the magnetic moment of the electron with the so-called Zitterbewegung. Later on many other models of the electron were proposed. On p.74 of his book “The Enigmatic Electron” Mac Gregor [54] lists more than thirty such models. At the end none of these models has been completely successful because the problem developed a seemingly insur- mountable difficulty when it was shown through electron-electron scattering experiments that the radius of the electron must be smaller than 10−16 cm, in other words that the electron appears to be a point particle, at least by three orders of magnitude smaller than the classical electron radius re = e2/mc2 = 2.8179·10−13 cm. This, of course, makes it very difficult to explain how a particle can have a finite angular momentum when the radius goes to zero, and how an electric charge can be confined in an infinitesimally small vol- ume. If the elementary electric charge would be in a volume with a radius of O(10−16) cm the Coulomb self-energy would be orders of magnitude larger than the rest mass of the electron, which is not realistic. The choice is be- tween a massless point charge and a finite size particle with a non-interacting mass to which an elementary electric charge is attached. It seems fair to say that at present, more than 100 years after the discovery of the electron, we do not have an accepted theoretical explanation of the electron. We propose in the following, as in [55], that the non-electromagnetic mass which seems to be necessary in order to explain the electron consists of neutrinos. This is actually a necessary consequence of our standing wave 39 model of the masses of the mesons and baryons. And we propose that the non-electromagnetic force required to hold the electric charge and the neu- trinos in the electron together is the weak nuclear force which, as we have suggested, holds together the masses of the mesons and baryons and also the mass of the muons. Since the range of the weak nuclear force is on the order of 10−16 cm the neutrinos must be arranged in a lattice with the weak force extending from each lattice point only to the nearest neighbors. The O(10−13) cm size of the neutrino lattice in the electron does not at all con- tradict the results of the scattering experiments, just as the explanation of the mass of the muons with the standing wave model does not contradict the apparent point particle", + "only to the nearest neighbors. The O(10−13) cm size of the neutrino lattice in the electron does not at all con- tradict the results of the scattering experiments, just as the explanation of the mass of the muons with the standing wave model does not contradict the apparent point particle characteristics of the muon, because neutrinos are in a very good approximation non-interacting and therefore are not noticed in scattering experiments with electrons. The rest mass of the electron is m(e) = 0.510 998 92 ± 4·10−8 MeV/c2 ⋆ and the electrostatic charge of the electron is e = 4.803 2044 · 10−10 esu, as stated in the Review of Particle Physics [2]. The objective of a theory of the electron must, first of all, be the explanation of m(e±) and e, but also of s(e±) and of the magnetic moment µe. We will first explain the rest mass of the electron making use of what we have learned from the standing wave model, in particular of what we have learned about the explanation of the mass of the µ± mesons in Section 7. The muons are leptons, just as the electrons, that means that they interact with other particles exclusively through the electric force. The muons have a mass which is 206.768 times larger than the mass of the electron, but they have the same elementary electric charge as the electron or positron and the same spin. Scattering experiments tell that the µ± mesons are point particles with a size < 10−16 cm, just as the electron. In other words, the muons have the same characteristics as the electrons and positrons but for a mass which is about 200 times larger. Consequently the muon is often referred to as a “heavy” electron. If a non-electromagnetic mass is required to explain the mass of the electron then a non-electromagnetic mass 200 times as large as in the electron is required to explain the mass of the muons. These non-electromagnetic masses must be non-interacting, otherwise scattering experiments could not find the size of either the electron or the muon at 10−16 cm. We have explained the mass of the muons with the standing wave model in Section 7. According to our model the muons consist of an elementary electric charge and an oscillating lattice of neutrinos. Neutrinos are the only non-interacting matter we know of. In the muon lattice are, according to 40 our model, (N - 1)/4 = N′/4 muon neutrinos νµ (respectively anti-muon neu- trinos ¯νµ), N′/4 electron neutrinos νe and the same number of anti-electron neutrinos ¯νe, one elementary electric charge and the energy of the lattice oscillations. The letter N stands for the number of all neutrinos and antineu- trinos in the cubic lattice of the π± mesons, N = 2.854·109, Eq.(15). It is, as explained in Section 7, a necessary consequence of the decay of, say, the µ− muon µ−→e−+ ¯νe + νµ that there must be N′/4 electron neutrinos νe in the emitted electron. For the mass of the electron neutrinos and anti-electron neutrinos we found in", + "= 2.854·109, Eq.(15). It is, as explained in Section 7, a necessary consequence of the decay of, say, the µ− muon µ−→e−+ ¯νe + νµ that there must be N′/4 electron neutrinos νe in the emitted electron. For the mass of the electron neutrinos and anti-electron neutrinos we found in Eq.(37) that m(νe) = m(¯νe) = 0.365 milli-eV/c2 ⋆. The sum of the energies in the rest masses of the N′/4 neutrinos or antineutrinos in the lattice of the electron or positron is then X m(νe)c2 ⋆= N′/4 · m(νe)c2 ⋆= 0.260 43 MeV = 0.5096 m(e±)c2 ⋆. (47) In other words, the sum of the rest masses of the neutrinos in the electron is approximately equal to 1/2 of the rest mass of the electron. The other half of the rest mass of the electron must originate from the energy in the electric charge carried by the electron. The explanation of the π± mesons led to the explanation of the µ± mesons and now leads to the explanation of the mass of e±. From pair production γ + M →e−+ e+ + M, (M being any nucleus), and from conservation of neutrino numbers follows necessarily that there must also be a neutrino lattice composed of N′/4 anti-electron neutrinos, which make up the lattice of the positrons, which lattice has, since m(νe) = m(¯νe), the same rest mass as the neutrino lattice of the electron, as it must be for the antiparticle of the electron. Conservation of charge requires conservation of neutrino numbers. Fourier analysis dictates that a continuum of high frequencies must be in the electrons or positrons created by pair production in a timespan of 10−23 seconds. We will now determine the energy Eν(e±) in the oscillations in the interior of the electron. Since we want to explain the rest mass of the electron we can only consider the frequencies of non-progressive waves, either standing waves or circular waves. The sum of the energies of the lattice oscillations is, in the case of the π± mesons, given by Eν(π±) = hν0N 2π(ehν/kT −1) π Z −π φ dφ . (48) This is Eq.(14) combined with Eq.(16) which were used to determine the 41 oscillation energy in the π0 and π± mesons. This equation was introduced by Born and v. Karman [13] in order to explain the internal energy of cubic crystals. If we apply Eq.(48) to the oscillations in the electron which has N′/4 electron neutrinos νe we arrive at Eν(e±) = 1/4·Eν(π±), which is mistaken because Eν(π±) ≈m(π±)c2 ⋆/2 and m(π±) = 273 m(e±). Eq.(48) must be modified in order to be suitable for the oscillations in the electron. It turns out that we must use Eν(e±) = hν0N · αf 2π(ehν/kT −1) π Z −π φ dφ , (49) where αf is the fine structure constant. As is well-known the fine structure constant αf characterizes the strength of the electromagnetic forces. The appearance of αf in Eq.(49) means that the nature of the oscillations in the electron is different from the oscillations in the π0 or π± lattices.", + "(49) where αf is the fine structure constant. As is well-known the fine structure constant αf characterizes the strength of the electromagnetic forces. The appearance of αf in Eq.(49) means that the nature of the oscillations in the electron is different from the oscillations in the π0 or π± lattices. With αf = e2/¯hc⋆and ν0 = c⋆/2πa we have hν0αf = e2/a , (50) which shows that the oscillations in the electron are electric oscillations. The appearance of e2 in Eq.(50) guarantees that the oscillation energy of the electron and positron are the same, as it must be. There must be N′/2 oscillations of the elements of the electric charge in e±, because we deal with non-progressive waves, which are the superposition of two waves. As we will see later the spin requires that the oscillations are circular. That means that 2×N′/4 ∼= N/2 oscillations are in Eq.(49). From Eqs.(48,49) then follows that Eν(e±) = αf/2 · Eν(π±) . (51) Eν(π±) is the oscillation energy in the π± mesons which can be calculated with Eq.(48). According to Eq.(28) it is Eν(π±) = 67.82 MeV = 0.486 m(π±)c2 ⋆≈m(π±)c2 ⋆/2 . (52) With Eν(π±) ≈m(π±)c2 ⋆/2 = 139.57/2 MeV and αf = 1/137.036 follows from Eq.(51) that the oscillation energy of the electron or positron is Eν(e±) = αf 2 · m(π±)c2 ⋆ 2 = 0.254 623 MeV = 0.996 570 m(e±)c2 ⋆/2 . (53) 42 If we replace in Eq.(53) the experimental value for m(π±) by the good em- pirical approximation m(π±) ∼= m(e±)(2/αf), Eq.(73), then it follows that Eν(e±) ∼= 1/2 · m(e±)c2 ⋆. (54) In other words, 1/2 of the energy in the rest mass of the electron is made up by the oscillation energy in e±. This equation corresponds to Eq.(29a) for the oscillation energy in the π± mesons. In Eq.(53) we have determined the value of the oscillation energy in e± from the product of the very accurately known fine structure constant and the very accurately known rest mass of the π± mesons which establishes a firm value of the oscillation energy of e±. The other half of the energy in m(e±) is in the rest masses of the neutrinos in the electron. We can confirm Eq.(53) without using Eν(π±) with the formula for the oscillation energy in the form of Eq.(58) with N/2 = 1.427·109, e = 4.803·10−10 esu, a = 1·10−16 cm, f(T) = 1/1.305·1013 and, with the integral being π2, we obtain Eν(e±) = 0.968 m(e±)c2 ⋆/2. This calculation involves more parameters than in Eq.(53) and is consequently less accurate than Eq.(53). In a good approximation the oscillation energy of e± in Eq.(53) is equal to the sum of the energies in the rest masses of the electron neutrinos in the e± lattice in Eq.(47), or Eν(e±) ∼= N′/4 · m(νe)c2. Since m(e±)c2 ⋆= Eν(e±) + X m(νe)c2 ⋆= Eν(e±) + N′/4 · m(νe)c2 ⋆, (55) it follows from Eqs.(47) and (53) that m(e±)c2 ⋆(theor) = 0.5151 MeV = 1.0079 m(e±)c2 ⋆(exp) . (56) The theoretical rest mass of the electron or positron agrees within", + "or Eν(e±) ∼= N′/4 · m(νe)c2. Since m(e±)c2 ⋆= Eν(e±) + X m(νe)c2 ⋆= Eν(e±) + N′/4 · m(νe)c2 ⋆, (55) it follows from Eqs.(47) and (53) that m(e±)c2 ⋆(theor) = 0.5151 MeV = 1.0079 m(e±)c2 ⋆(exp) . (56) The theoretical rest mass of the electron or positron agrees within the accu- racy of the parameters N and m(νe) with the measured rest mass. From Eq.(51) follows with Eν(π±) ∼= m(π±)c2 ⋆/2 from Eq.(52) that 2Eν(e±) = αfEν(π±) ∼= αfm(π±)c2 ⋆/2 ∼= m(e±)c2 ⋆, or that m(e±) · 2/αf = 274.072 m(e±) ∼= m(π±) , (57) whereas the actual ratio of the mass of the π± mesons to the mass of the elec- tron is m(π±)/m(e±) = 273.132 or 0.9965 · 2/αf. We have here the same ratio m(π±)/m(e±) which we find with the standing wave model of the π± mesons, 43 Eq.(73). This is a necessary condition for the validity of our model of the electron. We have thus shown that the rest mass of the electron or positron can be explained by the sum of the rest masses of the electron neutrinos or anti- electron neutrinos in a cubic lattice with N′/4 electron neutrinos νe plus the mass in the sum of the energy of N/2 standing electric oscillations in the lattice, Eq.(53). The one oscillation added to the even numbered N′/4 oscillations is the oscillation at the center of the lattice, Fig. 7. From this model follows, since it deals with a cubic neutrino lattice, that the electron is not a point particle. However, since neutrinos are non-interacting their presence will not be detected in electron-electron scattering experiments. Fig. 7: Horizontal or vertical section through the central part of the electron lattice. The size of the electron determined by electron-electron scattering at < 10−16 cm seems to contradict the model of the electron proposed here. However, this experimental size does not apply to the circumstances consid- ered here. One has to find the scattering formula for finite size charged cubic lattices and analyze the experimental data with such a scattering formula in order to see whether our model is in contradiction to the experiments. If the electron lattice behaves like a rigid body then electron-electron scattering should take place at the center of the lattice, which is a point [54]. Let us compare the rest mass of the electron to the rest mass of the 44 muon, which was explained in Section 7 with an oscillating lattice of muon and electron neutrinos and an elementary electric charge. The electron has the most simple neutrino lattice, consisting of only one neutrino type, ei- ther electron neutrinos or anti-electron neutrinos, and it has the smallest sum of the rest masses of the neutrinos in a particle. The heavy weight of the muon m(µ±) = 206.768 m(e±) is a consequence of the heavy weight of the N′/4 muon neutrinos or N′/4 anti-muon neutrinos in the muon lat- tice. The mass of either a muon neutrino or an anti-muon neutrino is 137 times the mass of an electron or anti-electron neutrino, Eq.(42). That", + "muon m(µ±) = 206.768 m(e±) is a consequence of the heavy weight of the N′/4 muon neutrinos or N′/4 anti-muon neutrinos in the muon lat- tice. The mass of either a muon neutrino or an anti-muon neutrino is 137 times the mass of an electron or anti-electron neutrino, Eq.(42). That makes the electron neutrinos and anti-electron neutrinos in µ± as well as the mass of the electric charge in a first approximation negligible. It then follows from Eq.(68) that m(µ±)(theor) ∼= 3/2 · m(νµ)/m(νe) · m(e±) = 205.11 m(e±) = 0.99198 m(µ±)(exp), which proves that the heavy mass of the muons is caused by the heavy muon neutrinos. In order to confirm the validity of our preceding explanation of the mass of the electron we must show that the sum of the charges in the electric oscillations in the interior of the electron is equal to the elementary electric charge of the electron. We recall that Fourier analysis requires that, after pair production, there must be a continuum of frequencies in the electron and positron. With hν0αf = e2/a from Eq.(50) follows from Eq.(49) that the oscillation energy in e± is the sum of 2×(N′/4 + 1) ∼= N/2 electric oscillations Eν(e±) = N 2 · e2 a · f(T) 2π π Z −π φ dφ , (58) with f(T) = 1/(ehν/kT−1) = 1/1.305·1013 from p.17. Inserting the values for N, f(T) and a we find that Eν(e±) = 0.968 m(e±)c2 ⋆/2 ∼= m(e±)c2 ⋆/2 as in Eq.(53). The discrepancy between m(e±)c2 ⋆/2 and Eν(e±) so calculated must originate from the uncertainty of the parameters N, f(T) and a in Eq.(58). In order to determine the charge e in the electric oscillations we replace the integral divided by 2π in Eq.(58), which has the value π/2, by the sum Σ φk∆φ, where k is an integer number with the maximal value km = (N/4)1/3. φk is equal to kπ/km and we have Σ φk ∆φ = km X k=1 kπ km · 1 km = km(km + 1)π 2 k2m ∼= π 2 , as it must be. The energy in the individual electric oscillation with index k 45 is then ∆Eν(k) = φk ∆φ = kπ/k2 m . (59) Suppose that the energy of the electric oscillations is correctly described by the self-energy of an electric charge U = 1/2 · Q2/r . (60) The self-energy of the elementary electric charge is normally used to deter- mine the mass of the electron from its charge, here we use Eq.(60) the other way around, we determine the charge from the energy in the oscillations. The charge of the electron is contained in the electric oscillations. That means that the electric charge of e± is not concentrated in a point, but is distributed over N/4 = O(109) charge elements Qk. The charge elements are distributed in a cubic lattice and the resulting electric field is cubic, not spherical. In the absence of a central force which originates at the center of the particle and affects all parts of the particle the configuration of", + "over N/4 = O(109) charge elements Qk. The charge elements are distributed in a cubic lattice and the resulting electric field is cubic, not spherical. In the absence of a central force which originates at the center of the particle and affects all parts of the particle the configuration of the particle is not spherical, just as it was with the shape of the π± mesons. For distances large as compared to the sidelength of the cube, (which is O(10−13) cm), say at the first Bohr radius which is on the order of 10−8 cm, the deviation of the cubic field from the spherical field will be reduced by about 10−10. The charge in all electric oscillations in the electron is Q = X k Qk . (61) Setting the radius r in the formula for the self-energy equal to 2 a we find, with Eqs.(58,59,60), that the charge in the individual electric oscillations is Qk = ± q 2π N e2f(T)/k2 m · √ k . (62) and with km = 1/2 · (N/4)1/3 = 447 and km X k=1 √ k = 6310.8 (63) follows, after we have doubled the sum over √ k, because for each index k there is a second oscillation on the negative axis of φ, that Q = Σ Qk = ± 5.027 · 10−10 esu , (64) 46 whereas the elementary electric charge is e = ± 4.803 · 10−10 esu. That means that our theoretical charge of the electron is 1.047 times the elementary elec- tric charge. Within the uncertainty of the parameters the theoretical charge of the electron agrees with the experimental charge e. We have confirmed that it follows from our explanation of the mass of the electron that the electron has, within a 5% error, the correct electric charge. Each element of the charge distribution is surrounded in the horizontal plane by four electron neutrinos as in Fig. 7, and in vertical direction by an electron neutrino above and also below the element. The electron neutrinos hold the charge elements in place. We must assume that the charge elements are bound to the neutrinos by the weak nuclear force. The weak nuclear force plays here a role similar to its role in holding, for example, the π± or µ± lattices together. It is not possible, in the absence of a definitive explanation of the neutrinos, to give an explanation for the electro-weak interaction between the electric oscillations and the neutrinos. However, the presence of the range a of the weak nuclear force in e2/a is a sign that the weak force is involved in the electric oscillations. The attraction of the charge elements by the neutrinos overcomes the Coulomb repulsion of the charge elements. The weak nuclear force is the missing non-electromagnetic force or the Poincar´e stress which holds the elementary electric charge together. The same considerations apply for the positive electric charge of the positron, only that then the electric oscillations are all of the positive sign and that they are bound to anti-electron neutrinos. Finally we learn that", + "missing non-electromagnetic force or the Poincar´e stress which holds the elementary electric charge together. The same considerations apply for the positive electric charge of the positron, only that then the electric oscillations are all of the positive sign and that they are bound to anti-electron neutrinos. Finally we learn that Eq.(58) precludes the possibility that the charge of the electron sits only on its surface. The number N in Eq.(58) would then be on the order of 106, whereas N must be on the order of 109 so that Eν(e±) can be m(e±)c2 ⋆/2 as is necessary. In other words, the charge of the electron must be distributed throughout the interior of the electron, as we postulated. Summing up: The rest mass of the electron or positron originates from the sum of the rest masses of N′/4 electron neutrinos or anti-electron neutrinos in cubic lattices plus the mass in the energy of the electric oscillations in their neutrino lattices. The neutrinos, as well as the electric oscillations, make up 1/2 of the rest mass of e± each. The electric oscillations are bound to the neutrinos by the weak nuclear force. The sum of the charge elements of the electric oscillations accounts for the elementary charge of the electron, respectively positron. The electron or the positron are not point particles. One hundred years of sophisticated theoretical work have made it abun- dantly clear that the electron is not a purely electromagnetic particle. There 47 must be something else in the electron but electric charge. It is equally clear from the most advanced scattering experiments that the “something else” in the electron must be non-interacting, otherwise it could not be that we find that the radius of the electron must be smaller than 10−16 cm. The only non-interacting matter we know of with certainty are the neutrinos. So it seems to be natural to ask whether neutrinos are not part of the electron. Actually we have not introduced the neutrinos in an axiomatic manner but rather as a consequence of our standing wave model of the stable mesons, baryons and µ mesons. It follows necessarily from this model that after the decay of, say, the µ−meson there must be electron neutrinos in the emitted electron, and that they make up one half of the rest mass of the electron. The other half of the energy in the electron originates from the energy of the electric oscillations. We have thus explained the rest masses of the electron or positron which agree, within 1% accuracy, with the experimental value of m(e±). We have learned that the charge of the electron is not concentrated in a single point, but rather is distributed over O(109) elements which are held together with the neutrinos by the weak nuclear force. The sum of the charges in the electric oscillations is, within the accuracy of the parameters, equal to the elementary charge of the electron. From the explanation of the mass and charge of the electron follows, as we will show, the correct spin and magnetic moment of the electron, the", + "The sum of the charges in the electric oscillations is, within the accuracy of the parameters, equal to the elementary charge of the electron. From the explanation of the mass and charge of the electron follows, as we will show, the correct spin and magnetic moment of the electron, the other two fundamental features of the electron. With a cubic lattice of anti-electron neutrinos we also arrive with the same considerations as above at the correct mass, charge, spin and magnetic moment of the positron. 10 The ratios m(µ±)/m(e±), m(π±)/m(e±) and m(p)/m(e) In order to check on the validity of our explanation of m(π±), m(µ±) and also of m(e±) we will now look at the ratios m(µ±)/m(e±), m(π±)/m(e±) and m(p)/m(e). In order to determine m(µ±)/m(e±) we first modify Eq.(39) by setting N′/4 · m(νe) = 0.5 m(e±), not at 0.51 m(e±). In other words we say that 1/2 of the mass of the electron is made of neutrinos. If the other half of the mass of the electron originates from the electric charge of the electron, as we have shown in the preceding Section, then the mass of the electron is twice the mass of the sum of the rest masses of the neutrinos in the electron 48 (Eq.39) and we have m(e±) = N′/2 · m(νe) or N′/2 · m(¯νe) . (65) We also set Eν(π±) = 0.5 m(π±)c2 ⋆, not at 0.486 m(π±)c2 ⋆as in Eq.(28). With Eν(π±) = Eν(µ±) from Eq.(33) follows with Eq.(31) and m(π±)c2 ⋆= 2 Eν(π±) that Eν(µ±) = N′/2 · [m(νµ) + m(νe)]c2 ⋆. (66) From Eq.(36) and with Eqs.(38,41) then follows that m(µ±) = 3/4 · N′m(νµ) + N′m(νe) , (67) considering only the neutrino masses, not the charge in µ±. With m(e±) = N′/2 · m(νe) from Eq.(65) we have m(µ±) m(e±) = 3 2 · m(νµ) m(νe) + 2 , (68) or with m(νµ)/m(νe) ∼= 1/αf from Eq.(42) it turns out that m(µ±) m(e±) ∼= 3 2 · 1 αf + 2 = 207.55 , (69) which is 1.0038 times the real ratio 206.768. In order to arrive at the proper ratio of m(µ±)/m(e±) the ratio of the neutrino masses m(νe)/m(νµ) must be equal to αf, as in Eq.(42). The mass of the muon is, according to Eq.(68), much larger than the mass of the electron because the mass of the muon neutrino is much larger, m(νµ) ∼= 137 m(νe), than the mass of the electron neutrino. The ratio of the mass of the muon to the mass of the electron is independent of the number N′ of the neutrinos in both lattices. Barut’s [41] empirical formula for the mass ratio is m(µ±)/m(e±) = 3/2αf + 1 = 206.55 , (70) whereas the actual ratio is 206.768 = 1.0010 · 206.55. A much better ap- proximation to the experimental mass ratio is obtained when the + 1 in Barut’s formula is replaced by + 1.25. The thus calculated m(µ±)/m(e±) = 206.804 differs then from the measured m(µ±)/m(e±) = 206.7683 by the factor 1.00017. 49 A better relation between the mass of the", + "much better ap- proximation to the experimental mass ratio is obtained when the + 1 in Barut’s formula is replaced by + 1.25. The thus calculated m(µ±)/m(e±) = 206.804 differs then from the measured m(µ±)/m(e±) = 206.7683 by the factor 1.00017. 49 A better relation between the mass of the muon and the mass of the pion can now be established. According to the third paragraph of Section 7 it is, empirically, m(µ±)/m(π±) = 3/4 + αf . (71) With αfEν(π±) = 2Eν(e±), (Eq.51), and m(π±)c2 ⋆= 2Eν(π±), as well as m(e±)c2 ⋆= 2Eν(e±), follows that m(µ±) ∼= 3/4 · m(π±) + 2 m(e±) . (72) The right hand side of Eq.(72) is 1.00039 times larger than the experimental value of m(µ±). The reason for the additional term 2m(e±) on the right hand side of Eq.(72) has still to be found. Similarly we obtain for the π± mesons the ratio m(π±) m(e±) = 2 [m(νµ) m(νe) + 1] ∼= 2 αf + 2 = 276.07 , (73) which is 1.0108 times the real ratio 273.1321. We have, however, only con- sidered the ratio of the rest masses of the neutrinos in π± and e±, not the consequences of either the charge or the spin in π± or e±. The empirical formula for the mass ratio m(π±)/m(e±) is m(π±)/m(e±) = 2/αf −1 = 273.07 , (74) which is 0.99977 times the experimental ratio 273.1320. The rest mass of the π± mesons is 273 ∼= 2 × 1/αf times larger than the rest mass of the electron because the muon or antimuon neutrino masses in π± are ∼= 137 times larger than the electron or anti-electron neutrino masses in the electron or positron. For the π0 meson we find empirically that m(π0)/m(e±) = 1.000 268 (2/α −10) = 264.1426 . (74a) The masses of the γ-branch particles are, according to Eq.(1), in a good approximation integer multiples of the mass of the π0 meson. Their masses are consequently multiples of the right hand side of Eq.(74a). For example we find that m(η)/m(e±)(exp) = 1.0144 · 4 · (2/α −10) and m(Λ)/m(e±)(exp) = 1.0335 · 8 · (2/α −10). In order to determine m(n)/m(e±) we start with K0 = (2.)π± + π∓and E((2.)π±) = 4Eν(π±) + N′/2 · [m(νµ) + m(νe)]c2 ⋆, Eq.(30). Then m(K0) = 50 7N′/2 · [m(νµ) + m(νe)], and with m(n) ∼= m(K0 + K0) = 2m(K0) follows that m(n) m(e±) = 14 [m(νµ) m(νe) + 1] = 1932.5 , (75) that is 105.2% of the experimental value 1836.15. The 5.2% excess of the calculated m(n)/m(e) is the consequence of the 2.6% excess of the theoretical m(K0) over the experimental m(K0). With m(p) = 0.9986 m(n) we have m(p) m(e) = 0.9986 · 14 [m(νµ) m(νe) + 1] ∼= 0.9986 [ 14 αf + 14] = 1929.79 , (76) that is 104.98% of the experimental ratio 1836.15. The empirical formula for m(p)/m(e) is m(p)/m(e) = 14 [1/αf −6] = 0.9980 m(p)/m(e)(exp) . (77) We arrive at a better agreement between the theoretical values of m(e±), m(µ±)/m(e±) and m(π±)/m(e±) and their actual", + "14 αf + 14] = 1929.79 , (76) that is 104.98% of the experimental ratio 1836.15. The empirical formula for m(p)/m(e) is m(p)/m(e) = 14 [1/αf −6] = 0.9980 m(p)/m(e)(exp) . (77) We arrive at a better agreement between the theoretical values of m(e±), m(µ±)/m(e±) and m(π±)/m(e±) and their actual values if we introduce a modified oscillation energy of the electron E′ ν(e±) given by E′ ν(e±) = Eν(e±)(1 + αf/2) . (78) With Eq.(53) we then have E′ ν(e±) = 1.000 206 m(e±)c 2 ⋆/2 . (79) The agreement of E′ ν(e±) given by Eq.(79) with the actual m(e±)c2 ⋆/2 is an order of magnitude better than it was with Eq.(53). The modification of the oscillation energy applies only to the electron, not to either µ± or π±, because according to Eq.(53) Eν(e±) is proportional to m(π±) and if m(π±) were also modified by (1 + αf/2) then E′ ν(π±) would be proportional to (1 + αf/2)2 and would be a worse approximation then Eq.(28). With m(e±)′ standing for the mass of the electron calculated with the modified oscillation energy E′ ν(e±), and assuming that m(e±)′c2 ⋆= 2E′ ν(e±), as seems to be the case according to Eq.(79), we obtain m(µ±) m(e±)′ = m(µ±) m(e±)(1 + αf/2) ∼= m(µ±) m(e±) (1 −αf/2) , (80) 51 and with Eq.(69) we have m(µ±) m(e±)′ = ( 3 2αf + 2)(1 −αf/2) = 3 2αf + 1.25 −αf , (81) which agrees remarkably well with the slightly modified form of Barut’s Eq.(70) in which the + 1 is replaced by + 1.25. So we have m(µ±) m(e±)′ = 206.7967 = 1.000 139 m(µ±) m(e±) (exp) . (82) Similarly we obtain for m(π±)/m(e±)′ with Eq.(51) m(π±) m(e±)′ = 2Eν(π±) 2E′ν(e±) ∼= 2Eν(π±)(1 −αf/2) αf/2 · 2Eν(π±) , (83) or m(π±) m(e±)′ = 2 αf (1 −αf/2) = 2 αf −1 = 273.072 , (84) whereas the experimental m(π±)/m(e±) is 273.132 = 1.000 2129 (2/αf −1). Our theoretical calculations of m(π±)/m(e±) and of m(µ±)/m(e±) agree within the percent range with their experimental values. This can only be if our explanation of m(π±), m(µ±) and m(e±) are correct in the same approx- imation, and if the ratio m(νe)/m(νµ) in Eq.(42) is valid. In other words, our theoretical values of m(π±)/m(e±) and of m(µ±)/m(e±) confirm the validity of our explanation of m(π±), m(µ±) and of m(e±), as well as the validity of the relation m(νe) = αfm(νµ). We have found that the leading term in the mass ratios, namely the term given by the ratio αf of the mass of the electron neutrino to the mass of the muon neutrino is, in our model, the same as in the empirical formulas for the mass ratios. On the other hand, there are differences on the right hand side of Eqs.(69,71) in the small second term of the mass ratios as compared to the same term in the empirical formulas for the mass ratios. The second term deals with the ratios of the number of the electron neutrinos in the mesons and baryons to the number of the elec- tron", + "Eqs.(69,71) in the small second term of the mass ratios as compared to the same term in the empirical formulas for the mass ratios. The second term deals with the ratios of the number of the electron neutrinos in the mesons and baryons to the number of the elec- tron neutrinos in the electron, without considering charge and spin. We have remedied these differences with the small modification of the oscillation en- ergy of the electron. The mass ratios m(µ±)/m(e±) and m(π±)/m(e±) then agree with the experimental mass ratios to the fourth decimal. The mass ratio of the proton to the electron cannot be remedied by the modification of the oscillation energy of the electron, because the difference between calcu- lated and experimental mass ratios is primarily caused by the 2.6 % excess of the theoretical rest mass of the K0 mesons. 52 11 The spin of the γ-branch particles It appears to be crucial for the validity of a model of the elementary particles that the model can also explain the spin of the particles without additional assumptions. The spin or the intrinsic angular momentum is, after the mass, the second most important property of the elementary particles. The stan- dard model does not explain the spin, the spin is imposed on the quarks. As is well-known the spin of the electron was discovered by Uhlenbeck and Goudsmit [50] more than 80 years ago. Later on it was established that the baryons have spin as well, but not the mesons. We have proposed an expla- nation of the spin of the particles in [56]. For current efforts to understand the spin of the nucleon see Jaffe [57] and of the spin structure of the Λ baryon see G¨ockeler et al. [58]. Rivas has described his own model of the spin and other spin models in his book [59]. The explanation of the spin requires an unambiguous answer, the spin must be 0 or 1/2 or integer multiples thereof, nothing else. For the explanation of the spin of the particles it seems to be necessary to have an explanation of the structure of the particles. The spin of a particle is, of course, the sum of the angular momentum vectors of the oscillations in the particle, plus the sum of the spin vectors of its neutrinos and antineutrinos, plus the spin of the electric charges which the particle carries. It is striking that the particles which, according to the standing wave model, consist of a single oscillation mode do not have spin, as the π0, π± and η mesons do, see Tables 1 and 2. It is also striking that particles whose mass is approximately twice the mass of a smaller particle have spin 1/2 as is the case with the Λ baryon, m(Λ) ≈2m(η), and with the nucleon m(n) ≈2m(K±) ≈2m(K0). The Ξ0 c baryon which is a doublet of one mode has also spin 1/2. Composite particles which consist of a doublet of one mode plus one or two other single modes have spin 1/2, as the Σ0, Ξ0 and", + "baryon, m(Λ) ≈2m(η), and with the nucleon m(n) ≈2m(K±) ≈2m(K0). The Ξ0 c baryon which is a doublet of one mode has also spin 1/2. Composite particles which consist of a doublet of one mode plus one or two other single modes have spin 1/2, as the Σ0, Ξ0 and Λ+ c , Σ0 c, Ω0 c baryons do. The only particle which seems to be the triplet of a single mode, the Ω−baryon with m(Ω−) ≈3m(η), has spin 3/2. It appears that the relation between the spin and the oscillation modes of the particles is straightforward. In the standing wave model the π0 and η mesons consist of N = 2.85·109 standing electromagnetic waves, each with its own frequency. Their oscilla- tions are longitudinal. The longitudinal oscillations of frequency νi in the π0 and η mesons do not have angular momentum or P i j(νi) = 0, with the in- dex running from 0 to N. Longitudinal oscillations cannot cause an intrinsic angular momentum because for longitudinal oscillations ⃗r × ⃗p = 0. 53 Each of the standing electromagnetic waves in the π0 and η mesons may, on the other hand, have spin s = 1 of its own, because circularly polarized electromagnetic waves have an angular momentum as was first suggested by Poynting [60] and verified by, among others, Allen [61]. The creation of the π0 meson in the reaction γ + p →π0 + p and conservation of angular momentum dictates that the sum of the angular momentum vectors of the N electromagnetic waves in the π0 meson is zero, P i j(si) = 0. Either the sum of the spin vectors of the electromagnetic waves in the π0 meson is zero, or each electromagnetic wave in the π0 meson has zero spin which would mean that they are linearly polarized. Linearly polarized electromagnetic waves are not expected to have angular momentum. That this is actually so was proven by Allen [61]. Since the longitudinal oscillations in the π0 and η mesons do not have angular momentum and since the sum of the spin vectors si of the electromagnetic waves is zero, the intrinsic angular momentum of the π0 and η mesons is zero, or j(π0, η) = X i j(νi) + X i j(si) = 0 (0 ≤i ≤N) . (85) In the standing wave model the π0 and η mesons do not have an intrinsic angular momentum or spin, as it must be. We now consider particles such as the Λ baryon whose mass is m(Λ) = 1.0190 · 2m(η). The Λ baryon seems to consist of the superposition of two oscillations of equal amplitudes and of frequencies ω and −ω, |−ω| = ω, at each of the N points of the lattice. The oscillations of such particles must be coupled what we have marked in Tables 1,2 by the ∗sign. The particles contain then N circular oscillations, each with its own frequency and each having an angular momentum of ¯h/2 as we will see. The superposition of two perpendicular linearly polarized traveling waves of equal amplitudes", + "particles must be coupled what we have marked in Tables 1,2 by the ∗sign. The particles contain then N circular oscillations, each with its own frequency and each having an angular momentum of ¯h/2 as we will see. The superposition of two perpendicular linearly polarized traveling waves of equal amplitudes and frequencies shifted in phase by π/2 leads to a circular wave with the constant angular momentum j = ¯h. As is well-known, the total energy of a traveling wave is the sum of the potential and the kinetic energy. In a traveling wave the kinetic energy is always equal to the potential energy. From Epot + Ekin = Etot = ¯hω , (86) follows Etot = 2Ekin = 2Θω2 2 = ¯hω, (87) 54 with the moment of inertia Θ. It follows that the angular momentum j is j = Θω = ¯h . (88) This applies to a traveling wave and corresponds to spin s = 1, or to a circularly polarized photon. We now add to one monochromatic circular oscillation with frequency ω a second circular oscillation with −ω of the same absolute value as ω but shifted in phase by π, having the same amplitude, as we have done in [56]. Negative frequencies are permitted solutions of the equations for the lattice oscillations, Eq.(7). In other words we consider the circular oscillations x(t) = exp[iωt] + exp[−i(ωt + π)] , (89) y(t) = exp[i(ωt + π/2)] + exp[−i(ωt + 3π/2)] . (90) This can also be written as x(t) = exp[iωt] −exp[−iωt] , (91) y(t) = i · (exp[iωt] + exp[−iωt]) . (92) If we replace i in the Eqs. above by −i we have a circular oscillation turning in opposite direction. The energy of the superposition of the two oscillations is the sum of the energies of both individual oscillations, and since in circular oscillations Ekin = Epot we have with Eq.(87) 4Ekin = 4Θω2/2 = Etot = ¯hω , (93) from which follows that the circular oscillation has an angular momentum j = Θω = ¯h/2 . (94) The superposition of two circular monochromatic oscillations of equal ampli- tudes and frequencies ω and −ω satisfies the necessary condition for spin s = 1/2 that the angular momentum is j = ¯h/2. The standing wave model treats the Λ baryon, which has spin s = 1/2 and a mass m(Λ) = 1.0190 · 2m(η), as the superposition of two particles of the same type with N standing electromagnetic waves. The waves are circular because they are the superposition of two waves with the same absolute 55 value of the frequency and the same amplitude. The angular momentum vectors of all circular waves in the lattice cancel, except for the wave at the center of the crystal. Each oscillation with frequency ω at φ > 0 has at its mirror position φ < 0 a wave with the frequency −ω, which has a negative angular momentum, since j = mr2ω and ω = ω0φ. Consequently the angular momentum vectors of both waves cancel. The center of the lattice", + "oscillation with frequency ω at φ > 0 has at its mirror position φ < 0 a wave with the frequency −ω, which has a negative angular momentum, since j = mr2ω and ω = ω0φ. Consequently the angular momentum vectors of both waves cancel. The center of the lattice oscillates, as all other lattice points do, but with the frequency ν(0) which is determined by the longest possible wavelength, which is twice the sidelength d of the lattice, so ν(0) = c/2d. As the other circular waves in the lattice the circular wave at the center has the angular momentum ¯h/2 according to Eq.(94). The angular momentum of the center wave is the only angular momentum which is not canceled by an oscillation of opposite circulation. There are, as it must be, three possible orientations of the axis of the circular oscillations. The net angular momentum of the N circular oscillations in the lattice which are superpositions of two oscillations reduces to the angular momentum of the center oscillation and is ¯h/2. Since the circular oscillations in the Λ baryon are the only possible contribution to an angular momentum the intrinsic angular momentum of the Λ baryon is ¯h/2 or j(Λ) = X i j(ωi) = j(ω0) = ¯h/2 . (95) We have thus explained that the Λ and likewise the Ξ0 c baryon satisfy the necessary condition that j = ¯h/2 for s = 1/2. The intrinsic angular momen- tum of the Λ baryon is the consequence of the superposition of two circular oscillations of the same amplitude and the same absolute value of the fre- quency. Spin 1/2 is caused by the composition of the particles, it is not a contributor to the mass of a particle. The other particles of the γ-branch, the Σ0, Ξ0, Λ+ c , Σ0 c and Ω0 c baryons are composites of a baryon with spin 1/2 plus one or two π mesons which do not have spin. Consequently the spin of these particles is 1/2. The spin of all particles of the γ-branch, exempting the spin of the Ω−baryon, has thus been explained. For an explanation of s(Σ±,0) = 1/2 and of s(Ξ−,0) = 1/2, regardless whether the particles are charged or neutral, we refer to [56]. 12 The spin of the particles of the ν-branch The characteristic particles of the neutrino-branch are the π± mesons which have zero spin. At first glance it seems to be odd that the π± mesons do not 56 have spin, because it seems that the π± mesons should have spin 1/2 from the spin of the charges e± in π±. What happens to the spin of e± in π± ? The solution of this puzzle is in the composition of the π± mesons which are, according to the standing wave model, made of a lattice of neutrinos and antineutrinos (Fig. 2) each having spin 1/2, the lattice oscillations, and an elementary electric charge. The longitudinal oscillations in the neutrino lattice of the π± mesons do not cause an angular momentum, P i j(νi) = 0,", + "according to the standing wave model, made of a lattice of neutrinos and antineutrinos (Fig. 2) each having spin 1/2, the lattice oscillations, and an elementary electric charge. The longitudinal oscillations in the neutrino lattice of the π± mesons do not cause an angular momentum, P i j(νi) = 0, as it was with the π0 meson. In the cubic lattice of N = O(109) neutrinos and antineutrinos of the π± me- sons the spin of nearly all neutrinos and antineutrinos must cancel because conservation of angular momentum during the creation of the particle re- quires that the total angular momentum around a central axis is ¯h/2. In fact the spin vectors of all but the neutrino or antineutrino in the center of the lattice cancel. In order for this to be so the spin vector of any particular neutrino in the lattice has to be opposite to the spin vector of the neutrino at its mirror position. As is well-known only left-handed neutrinos and right- handed antineutrinos exist. From ν = ν0φ (Eq.13) follows that the direction of motion of the neutrinos in e.g. the upper right quadrant (φ > 0) is opposite to the direction of motion in the lower left quadrant (φ < 0). Consequently the spin vectors of all neutrinos or antineutrinos in opposite quadrants are opposite and cancel. The only angular momentum remaining from the spin of the neutrinos of the lattice is the angular momentum of the neutrino or antineutrino at the center of the lattice which does not have a mirror parti- cle. Consequently the electrically neutral neutrino lattice consisting of N′/2 neutrinos and N′/2 antineutrinos and the center particle, each with spin j(ni) = 1/2, has an intrinsic angular momentum j = P i j(ni) = j(n0) = ¯h/2. But electrons or positrons added to the neutral neutrino lattice have likewise spin 1/2. If the spin of the electron or positron added to the neutrino lattice is opposite to the spin of the neutrino or antineutrino in the center of the lattice then the net spin of the π+ or π−mesons is zero, or j(π±) = X i j(νi)+ X i j(ni)+j(e±) = j(n0)+j(e±) = 0 (0 ≤i ≤N) . (96) It is important for the understanding of the structure of the π± mesons to realize that s(π±) = 0 can only be explained if the π± mesons consist of a neutrino lattice to which an electron or positron is added whose spin is opposite to the net spin of the neutrino lattice. Spin 1/2 of the elementary electric charge can only be canceled by something that has also spin 1/2, and 57 the only conventional choice for that is a single neutrino. The spin, the mass and the decay of π± require that the π± mesons are made of a cubic neutrino lattice and an elementary electric charge e±. The spin of the K± mesons is zero. With the spin of the K± mesons we encounter the same oddity we have just observed with the spin of the π± mesons, namely we have", + "the π± mesons are made of a cubic neutrino lattice and an elementary electric charge e±. The spin of the K± mesons is zero. With the spin of the K± mesons we encounter the same oddity we have just observed with the spin of the π± mesons, namely we have a particle which carries an elementary electric charge with spin 1/2, and nevertheless the particle does not have spin. The explanation of s(K±) = 0 follows the same lines as the explanation of the spin of the π± mesons. In the standing wave model the K± mesons are de- scribed by the state (2.)π± + π0, that means by the second mode of the lattice oscillations of the π± mesons plus a π0 meson. The second mode of the longitudinal oscillations of a neutral neutrino lattice does not have a net intrinsic angular momentum P i j(νi) = 0. But the spin of the neutrinos contributes an angular momentum ¯h/2, which originates from the neutrino or antineutrino in the center of the lattice, just as it is with the neutrino lattice in the π± mesons, so P i j(ni) = j(n0) = ¯h/2. Adding an elementary electric charge with a spin opposite to the net intrinsic angular momentum of the neutrino lattice creates the charged (2.)π± mode which has zero spin j((2.)π±) = X i j(ni) + j(e±) = j(n0) + j(e±) = 0 . (97) As discussed in Section 6 it is necessary to add a π0 meson to the second mode of the π± mesons in order to obtain the correct mass and the correct decays of the K± mesons. Since the π0 meson does not have spin the addition of the π0 meson does not add to the intrinsic angular momentum of the K± mesons. So s(K±) = 0 as it must be. The explanation of s = 0 of the K0 and K0 mesons described by the state (2.)π± + π∓is different because there is now no electric charge whose spin canceled the spin of the neutrino lattice in π±. The longitudinal oscillations of the second mode of the neutrino oscillations of π± in K0 as well as of the basic π∓mode do not create an angular momentum, P i j(νi) = 0. The sum of the spin vectors of the neutrinos in K0 and K0 is determined by the neutrinos in the second mode of the π± mesons, or the (2.)π± state, and the basic π∓mode, each have N′/2 neutrinos and N′/2 antineutrinos plus a center neutrino or antineutrino, so the number of all neutrinos and antineutrinos in the sum of both states, the K0,K0 mesons, is 2N. Since the size of the lattice of the K± mesons and the K0 mesons is the same it follows that two neutrinos are at each lattice point of the K0 or K0 mesons. We assume that Pauli’s 58 exclusion principle applies to neutrinos as well. Consequently each neutrino at each lattice point must share its location with an antineutrino. That means that the contribution of the spin of all", + "that two neutrinos are at each lattice point of the K0 or K0 mesons. We assume that Pauli’s 58 exclusion principle applies to neutrinos as well. Consequently each neutrino at each lattice point must share its location with an antineutrino. That means that the contribution of the spin of all neutrinos and antineutrinos to the intrinsic angular momentum of the K0 meson is zero or P i j(2ni) = 0. The sum of the spin vectors of the two opposite charges in either the K0 or the K0 mesons, or in the (2.)π± + π∓state, is also zero. Since neither the lattice oscillations nor the spin of the neutrinos and antineutrinos nor the electric charges contribute an angular momentum j(K0) = X i j(νi) + X i j(2ni) + j(e+ + e−) = 0 . (98) The intrinsic angular momentum of the K0 and K0 mesons is zero, or s(K0,K0) = 0, as it must be. In simple terms since, e.g., the structure of K0 is (2.)π+ + π−, the spin of K0 is the sum of the spin of (2.)π+ and of π−, both of which do not have spin. It does not seem possible to arrive at s(K0,K0) = 0 if both particles do not contain the N pairs of neutrinos and antineutrinos required by the (2.)π± + π∓state which we have suggested in Section 6. In the case of the neutron one must wonder how it comes about that a particle which seems to be the superposition of two particles without spin ends up with spin 1/2. The neutron, which has a mass ≈2m(K±) or 2m(K0), is either the superposition of a K+ and a K−meson or of a K0 and a K0 meson. The intrinsic angular momentum of the superposition of K+ and K−is either 0 or ¯h, which means that the neutron cannot be the superposition of K+ and K−. For a proof of this statement we refer to [56]. On the other hand the neutron can be the superposition of a K0 and a K0 meson. A significant change in the lattice occurs when a K0 and a K0 meson are superposed. Since each K0 meson contains N neutrinos and N an- tineutrinos, as we discussed in context with the spin of K0, the number of all neutrinos and antineutrinos in superposed K0 and K0 lattices is 4N. Since the size of the lattice of the proton as well of the neutron is the same as the size of K0 each of the N lattice points of the neutron now contains four neutrinos, a muon neutrino and an anti-muon neutrino as well as an electron neutrino and an anti-electron neutrino. The νµ, ¯νµ, νe, ¯νe quadrupoles oscil- late just like individual neutrinos do because we learned from Eq.(7) that the ratios of the oscillation frequencies are independent of the mass as well as of the interaction constant between the lattice points. In the neutrino quadrupoles the spin of the neutrinos and antineutrinos cancels, P i j(4ni) = 0. The superposition of two circular neutrino lattice oscillations,", + "from Eq.(7) that the ratios of the oscillation frequencies are independent of the mass as well as of the interaction constant between the lattice points. In the neutrino quadrupoles the spin of the neutrinos and antineutrinos cancels, P i j(4ni) = 0. The superposition of two circular neutrino lattice oscillations, that means 59 the circular oscillations of frequency ωi, contribute as before the angular mo- mentum of the center circular oscillation, so P i j(ωi) = j(ω0) = ¯h/2. The spin and charge of the four electrical charges e+e−e+e−hidden in the sum of the K0 and K0 mesons cancel, j(4e±) = 0. It follows that the intrinsic angular momentum of a neutron created by the superposition of a K0 and a K0 meson comes from the circular neutrino lattice oscillations only and is j(n) = X i j(ωi) + X i j(4ni) + j(4e±) = X i j(ωi) = j(ω0) = ¯h/2 , (99) as it must be. In simple terms, the spin of the neutron originates from the superposition of two circular neutrino lattice oscillations with the frequencies ω and −ω shifted in phase by π, which produces the angular momentum ¯h/2 at the center. The spin of the proton is 1/2 and is unambiguously defined by the decay of the neutron n →p + e−+ ¯νe. We have suggested in Section 7 that 3/4·N′ anti-electron neutrinos of the neutrino lattice of the neutron are removed in the β-decay of the neutron and that the other N′/4 anti-electron neutrinos leave with the emitted electron. The intrinsic angular momentum of the proton originates then from the spin of the central νµ¯νµνe triplet, from the spin of the e+e−e+ triplet which is part of the remains of the neutron, and from the angular momentum of the center of the lattice oscillations with the superposition of two circular oscillations. The spin of the central νµ¯νµνe triplet is canceled by the spin of the e+e−e+ triplet. According to the standing wave model the intrinsic angular momentum of the proton is j(p) = j(νµ¯νµνe)0 + j(e+e−e+) + j(ω0) = j(ω0) = ¯h/2 , (100) as it must be. The other mesons of the neutrino branch, the D±,0 and D± s mesons, both having zero spin, are superpositions of a proton and an antineutron of op- posite spin, or of their antiparticles, or of a neutron and an antineutron of opposite spin in D0. The spin of D± and D0 does therefore not pose a new problem. For an explanation of the spin of µ± we refer to [62]. Since all muon or anti-muon neutrinos have been removed from the π± lattice in the π± decay it follows that a neutrino vacancy is at the center of the µ± lattice (Fig. 6). Without a neutrino in the center of the lattice the sum of the spin vectors of all neutrinos in the µ± lattice is zero. However the µ± mesons consist of the 60 neutrino lattice plus an electric charge e± whose spin is 1/2. The spin of the µ± mesons originates from the spin of the", + "center of the lattice the sum of the spin vectors of all neutrinos in the µ± lattice is zero. However the µ± mesons consist of the 60 neutrino lattice plus an electric charge e± whose spin is 1/2. The spin of the µ± mesons originates from the spin of the elementary electric charge carried by the µ± mesons and is consequently s(µ±) = 1/2. The same considerations apply for the spin of τ ±, s(τ ±) = 1/2. An explanation of the spin of the mesons and baryons can only be valid if the same explanation also applies to the antiparticles of these particles whose spin is the same as that of the ordinary particles. The antiparticles of the γ-branch consist of electromagnetic waves whose frequencies differ from the frequencies of the ordinary particles only by their sign. The angular mo- mentum of the superposition of two circular oscillations with −ω and ω has the same angular momentum as the superposition of two circular oscillations with frequencies of opposite sign, as in Λ. Consequently the spin of the an- tiparticles of the γ-branch is the same as the spin of the ordinary particles of the γ-branch. The same considerations apply to the circular neutrino lattice oscillations which cause the spin of the neutron, the only particle of the ν- branch which has spin. In the standing wave model the spin of the neutron and the antineutron is the same. Let us summarize: The spin of the particles of the ν-branch originates from the angular momentum vectors of the oscillations and the spin vectors of the neutrinos in the particles and the spin vector of the elementary elec- tric charge or charges a particle carries. The contribution of all or all but one of the O(109) oscillations and O(109) neutrinos to the intrinsic angular momentum of the particles must cancel, otherwise the spin cannot be either 0 or 1/2. It requires the symmetry of a cubic lattice for this to happen. The center of the lattices alone determines the intrinsic angular momentum of the oscillations and neutrinos in the lattice. Adding to that the spin vector of one (or more) elementary electric charges with spin 1/2 and we arrive at the total intrinsic angular momentum of a particle. The most illuminating case are the π± mesons which do not have spin although they carry an elementary electric charge. Actually the neutrino lattice of the π± mesons has spin 1/2 from its central neutrino, but this spin vector is canceled by the spin of the electric charge e±, so s(π±) = 0. From the foregoing we arrive also at an understanding of the reason for the astonishing fact that the intrinsic angular momentum or spin of the par- ticles is independent of the mass of the particles, as exemplified by the spin ¯h/2 of the electron being the same as the spin ¯h/2 of the proton, notwith- standing the fact that the mass of the proton is 1836 times larger than the mass of the electron. However, in our model, the spin of the", + "of the particles, as exemplified by the spin ¯h/2 of the electron being the same as the spin ¯h/2 of the proton, notwith- standing the fact that the mass of the proton is 1836 times larger than the mass of the electron. However, in our model, the spin of the particles includ- 61 ing the electron is determined solely by the angular momentum ¯h/2 at the center of the lattice, the other angular momentum vectors in the particles cancel. The spin is independent of the number of the lattice points in a cubic lattice. Hence the mass of the particles in the other 109 lattice points is inconsequential for the intrinsic angular momentum of the particles. In our model the spin of the particles is independent of the mass of the particles, as it must be. 13 The spin and magnetic moment of the electron The model of the electron we have proposed in Section 9 has, in order to be valid, to pass a crucial test; the model has to explain satisfactorily the spin and the magnetic moment of the electron. When Uhlenbeck and Goudsmit [50] (U&G) discovered the existence of the spin of the electron they also pro- posed that the electron has a magnetic moment with a value equal to Bohr’s magnetic moment µB = e¯h/2m(e±)c⋆. Bohr’s magnetic moment results from the motion of an electron on a circular orbit around a proton. The magnetic moment of the electron postulated by U&G has been confirmed experimen- tally, but has been corrected by about 0.11% for the so-called anomalous magnetic moment. If one tries to explain the magnetic moment of the elec- tron with an electric charge moving on a circular orbit around the particle center, analogous to the magnetic moment of hydrogen, one ends up with ve- locities larger than the velocity of light, which cannot be, as already noted by U&G. It remains to be explained how the magnetic moment of the electron comes about. We will have to explain the spin of the electron first. The spin, or the intrinsic angular momentum of a particle is, of course, the sum of the angular momentum vectors of all components of the particle. In the electron these are the neutrinos and the electric oscillations. Each neutrino has spin 1/2 and in order for the electron to have s = 1/2 all, or all but one, of the spin vectors of the neutrinos in their lattice must cancel. If the neutrinos are in a simple cubic lattice as in Fig. 7 and the center particle of the lattice is not a neutrino, as in Fig. 7, the spin vectors of all neutrinos cancel, Σ j(ni) = 0, provided that the spin vectors of the neutrinos of the lattice point in opposite direction at their mirror points in the lattice. Otherwise the spin vectors of 62 the neutrinos would add up and make a very large angular momentum. We follow here the procedure we used in [56] to explain the spin of the muons. The spin vectors of all electron neutrinos", + "opposite direction at their mirror points in the lattice. Otherwise the spin vectors of 62 the neutrinos would add up and make a very large angular momentum. We follow here the procedure we used in [56] to explain the spin of the muons. The spin vectors of all electron neutrinos in the electron cancel, just as the spin vectors of all muon and electron neutrinos in the muons cancel, because there is a neutrino vacancy at the center of their lattices, Figs. 6,7. We will now see whether the electric oscillations in the electron contribute to its angular momentum. As we said in context with Eq.(51) there must be two times as many electric oscillations in the electron lattice than there are neutrinos. The oscillations can either be standing waves consisting of two linear oscillations or two circular oscillations with the frequencies ω and −ω. Both the standing waves and the circular oscillations are non-progressive and can be part of the rest mass of a particle. We will now assume that the electric oscillation are circular. Circular oscillations have an angular momentum ⃗j = m⃗r × ⃗v. And, as in the case of the spin vectors of the neutrinos, all or all but one of the O(109) angular momentum vectors of the electric oscillations must cancel in order for the electron to have spin 1/2. We describe the superposition of the two oscillations by x(t) = exp[iωt] + exp[−i(ωt + π)] , (101) y(t) = exp[i(ωt + π/2)] + exp[−i(ωt + 3π/2)] , (102) that means by the superposition of an oscillation with the frequency ω and a second oscillation with the frequency −ω as in Eqs.(89,90). The oscilla- tion with −ω is shifted in phase by π. Negative frequencies are permitted solutions of the equations of motion in a cubic lattice, Eqs.(7,13). As is well- known oscillating electric charges should emit radiation. However, this rule does already not hold in the hydrogen atom, so we will assume that the rule does not hold within the electron either. In circular oscillations the kinetic energy is always equal to the potential energy and the sum of both is the total energy. From Epot + Ekin = 2 Ekin = Etot (103) follows, as discussed in Eqs.(93,94), that the angular momentum of the su- perposition of the two circular oscillations is j = Θ ω = ¯h/2 . (104) 63 That means that each of the O(109) superposed circular electric oscillations has an angular momentum ¯h/2. The circulation of the circular oscillations in Eqs.(101,102) is opposite for all φ of opposite sign. It follows from the equation for the displacements un of the lattice points Eq.(5) un = Aei(ω t + nφ) , (105) that the velocities of the lattice points are given by vn = ˙un = i ωn un . (106) The sign of ωn changes with the sign of φ because the frequencies are given by Eq.(13), that means by ωn = ± ω0( φn + φ0 ) . (107) Consequently the circulation of the electric oscillations is opposite to the", + "vn = ˙un = i ωn un . (106) The sign of ωn changes with the sign of φ because the frequencies are given by Eq.(13), that means by ωn = ± ω0( φn + φ0 ) . (107) Consequently the circulation of the electric oscillations is opposite to the circulation at the mirror points in the lattice and the angular momentum vectors cancel, but for the angular momentum vector of the electric oscilla- tion at the center of the lattice. The center circular oscillation has, as all other electric oscillations, the angular momentum ¯h/2 as Eq.(104) says. The angular momentum of the entire electron lattice is therefore j(e±) = X i j(ni) + X i j(eli) = j(el0) = ¯h/2 , (108) as it must be for spin s = 1/2. The explanation of the spin of the electron given here follows the explanation of the spin of the baryons, as well as the explanation of the absence of spin in the mesons. A valid explanation of the spin must be applicable to all particles, in particular to the electron, the prototype of a particle with spin. We will now turn to the magnetic moment of the electron which is known with extraordinary accuracy, µe = 1.001 159 652 186 µB, according to the Re- view of Particle Physics [2], µB being Bohr’s magneton. The decimals after 1.00 µB are caused by the anomalous magnetic moment which we will not consider. As is well-known the magnetic dipole moment of a particle with spin is, in Gaussian units, given by ⃗µ = g e¯h 2mc⋆ ⃗s , (109) 64 where g is the dimensionless Land´e factor, m the rest mass of the particle that carries the charge e and ⃗s the spin vector. The g-factor has been intro- duced in order to bring the magnetic moment of the electron into agreement with the experimental facts. As U&G postulated and as has been confirmed experimentally the g-factor of the electron is 2. With the spin s = 1/2 and g = 2 the magnetic dipole moment of the electron is then µe = e¯h/2m(e±)c⋆, (110) or one Bohr magneton µB in agreement with the experiments, neglecting the anomalous moment. For a structureless point particle Dirac [51] has explained why g = 2 for the electron. However we consider here an electron with a finite size and which is at rest. When it is at rest the electron has still its magnetic moment. Dirac’s theory does therefore not apply here. The only part of Eq.(110) that can be changed in order to explain the g-factor of an electron with structure is the ratio e/m which deals with the spatial distribution of charge and mass. If part of the mass of the electron is non-electromagnetic and the non-electromagnetic part of the mass does not contribute to the magnetic moment of the electron, which to all that we know is true for neutrinos, then the ratio e/m in Eq.(109) is not e/m(e±) in the case of the electron. The elementary charge e certainly remains un- changed,", + "and the non-electromagnetic part of the mass does not contribute to the magnetic moment of the electron, which to all that we know is true for neutrinos, then the ratio e/m in Eq.(109) is not e/m(e±) in the case of the electron. The elementary charge e certainly remains un- changed, but e/m depends on what fraction of the mass of the electron is of electromagnetic origin and what fraction of the mass is non-electromagnetic. Only the current, not the mass of a current loop, determines the magnetic moment of a loop. From the very accurately known values of αf, m(π±)c2 ⋆ and m(e±)c2 ⋆and from Eq.(53) for the energy in the electric oscillations in the electron Eν(e±) = αf/2 · m(π±)c2 ⋆/2 = 0.996570 m(e±)c2 ⋆/2 follows that very nearly one half of the mass of the electron is of electric origin, the other half of m(e±) is made of neutrinos, Eq.(47), and neutrinos do not contribute to the magnetic moment. That means that in the electron the mass that carries the charge e is approximately m(e±)/2. The magnetic moment of the electron is then ⃗µe = g e¯h 2m(e±)/2 · c⋆ ⃗s , (111) and with s = 1/2 we have µe = g e¯h/2m(e±)c⋆. Because of Eq.(110) the g-factor must be equal to one and is unnecessary. In other words, if exactly one half of the mass of the electron consists of neutrinos, then it follows automatically that the electron has the correct magnetic moment µe = e¯h/2m(e±)c⋆without the artificial g-factor. 65 Conclusions We conclude that the standing wave model solves a number of problems for which an answer heretofore has been hard to come by. From the creation of the π0 meson and its decay into γγ after 10−16 sec follows that the π0 meson and the other particles of the γ-branch must consist of standing electromag- netic waves. The rest masses of the particles of the γ-branch obey the integer multiple rule. From the creation of the π± mesons and their decay follows that the π± mesons and the other particles of the ν-branch must consist of a lattice of νµ, ¯νµ, νe, ¯νe neutrinos, their oscillation energies and one or more elementary electric charges. From the explanation of the π± mesons follows that the rest mass of the µ± mesons is ∼= 3/4 of the rest mass of the π± mesons and it also follows that 1/2 of the rest mass of the electron or positron must consist of electron neutrinos or anti-electron neutrinos. The other half of the rest mass of the electron is in the energy of electric oscillations. The elemen- tary electric charge of the electron is the sum of the charges carried by the individual electric oscillations. The rest mass of the µ± mesons is so much larger than the rest mass of the electron or positron because the rest mass of the muon neutrinos is so much larger than the rest mass of the electron or anti-electron neutrinos. We have verified the validity of our explanation of m(e±), m(µ±) and m(π±) and", + "mesons is so much larger than the rest mass of the electron or positron because the rest mass of the muon neutrinos is so much larger than the rest mass of the electron or anti-electron neutrinos. We have verified the validity of our explanation of m(e±), m(µ±) and m(π±) and of the relation m(νe) = αf · m(νµ) by showing that the calculated ratios m(µ±)/m(e±) and m(π±)/m(e±) agree within 1% with the experimental values of these mass ratios. Only photons, neutrinos, charge and the weak nuclear force are needed to explain the masses of the electron, the muon and of the stable mesons and baryons and their antipar- ticles. We can explain the spin of the baryons and the absence of spin in the mesons, and the spin of the electron and muon as well; without making additional assumptions. We have also determined the rest masses of the e, µ and τ neutrinos and found that the mass of the electron neutrino is equal to the mass of the muon neutrino times the fine structure constant. Acknowledgments. I gratefully acknowledge the contributions of Dr. T. Koschmieder to this study. I thank Professor A. Gsponer for information about the history of the integer multiple rule. 66 References [1] Gell-Mann,M. 1964. Phys.Lett.B 111,1. [2] Yao,W.-M. et al. 2006. J.Phys.G 33,1. [3] El Naschie,M.S. 2002. Chaos,Sol.Frac. 14,649. [4] El Naschie,M.S. 2002. Chaos,Sol.Frac. 14,369. [5] El Naschie,M.S. 2003. Chaos,Sol.Frac. 16,353. [6] El Naschie,M.S. 2003. Chaos,Sol.Frac. 16,637. [7] Feynman,R.P. 1985. QED. The Strange Theory of Light and Matter. Princeton University Press, p.152. [8] Koschmieder,E.L. 2003. Chaos,Sol.Frac. 18,1129. [9] Nambu,Y. 1952. Prog.Th.Phys. 7,595. [10] Fr¨ohlich,H. 1958. Nucl.Phys. 7,148. [11] Koschmieder,E.L. and Koschmieder,T.H. 1999. 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Acad.Roy.Belgique, X,281,1999; arXiv.org/hep-ph/0002179; 2000) we relied on the data available at that time, Barnett et al. Rev.Mod.Phys. 68,611,1996. Since then the data on the particle masses have multiplied and become more precise. It seems now that particles with masses even 70 × m(π0) obey the integer multiple rule. A list of the particles which follow the integer multiple rule is given in Table 3. The masses are taken from the Review of Particle Physics [2]. Only par- ticles, stable or unstable, with J ≤1 are listed. To be on the safe side, we use only particles which the Particle Data Group considers to be “established”. The Ω−baryon with J = 3/2 is also given for comparison, but is not included in the least square analysis. The correlation of the masses of the different η mesons has also been in- vestigated by Palazzi, , using a mass unit of 33.86 MeV/c2, which is 1.0034 m(π0)/4. In his Fig. 2a he 69 Table 3: The particles following the integer multiple rule J m/m(π0) multiples J m/m(π0) multiples π0 0 1.0000 1.0000 · 1π0 ηc(1S) 0 22.0809 1.0037 · 22π0 η 0 4.0559 1.0140 · 4π0 J/ψ 1 22.9441 0.9976 ·23π0 η′(958) 0 7.0959 1.0137 · 7π0 χc0(1P) 0 25.2989 1.0119 · 25π0 η(1295) 0 9.5868 0.9587 · 10π0 χc1(1P) 1 26.0094 1.0004 · 26π0 η(1405) 0 10.445 1.0445 · 10π0 ηc(2S) 0 26.9528 0.9983 · 27π0 η(1475) 0 10.9352 0.9941 · 11π0 ψ(2S) 1 27.3091 1.0115 · 27π0 ψ(3770) 1 27.9389 0.9978 · 28π0 Λ 1/2 8.2658 1.0332 · 8π0 ψ(4040) 1 29.9237 0.9975 · 30π0 Λ(1405) 1/2 10.4166 1.0417 · 10π0 ψ(4191) 1 31.0543 1.00175 · 31π0 Λ(1670) 1/2 12.3725 1.0310 · 12π0 ψ(4415) 1 32.7538 0.9925 · 33π0 Λ(1800) 1/2", + "ψ(2S) 1 27.3091 1.0115 · 27π0 ψ(3770) 1 27.9389 0.9978 · 28π0 Λ 1/2 8.2658 1.0332 · 8π0 ψ(4040) 1 29.9237 0.9975 · 30π0 Λ(1405) 1/2 10.4166 1.0417 · 10π0 ψ(4191) 1 31.0543 1.00175 · 31π0 Λ(1670) 1/2 12.3725 1.0310 · 12π0 ψ(4415) 1 32.7538 0.9925 · 33π0 Λ(1800) 1/2 13.335 1.0258 · 13π0 Σ0 1/2 8.8359 0.9818 · 9π0 Υ(1S) 1 70.0884 1.0013 · 70π0 Σ(1660) 1/2 12.2984 1.0249 · 12π0 χb0(1P) 0 73.0455 1.0006 · 73π0 Σ(1750) 1/2 12.9652 0.9973 · 13π0 χb1(1P) 1 73.2926 1.0040 · 73π0 Ξ0 1/2 9.7412 0.9741 · 10π0 Υ(2S) 1 75.8102 0.9975 · 76π0 Ω− 3/2 12.3907 1.0326 · 12π0 χb0(2P) 0 75.8094 0.9975 · 76π0 Λ+ c 1/2 16.9397 0.9965 · 17π0 χb1(2P) 1 75.9795 0.9997 · 76π0 Λc(2593)+ 1/2 19.2285 1.0120 · 19π0 Υ(3S) 1 76.7185 0.9963 · 77π0 Σc(2455)0 1/2 18.1792 1.00995 · 18π0 Υ(4S) 1 78.3795 1.0049 · 78π0 Ξ0 c 1/2 18.3069 1.01705 · 18π0 Υ(10860) 1 80.4954 1.0062 · 80π0 Ξ′0 c 1/2 19.0996 1.0052 ·19π0 Υ(11020) 1 81.6364 0.9956 · 82π0 Ξc(2790) 1/2 20.6843 0.9850 · 21π0 Ω0 c 1/2 19.9849 0.9992 · 20π0 70 shows that the masses of the η mesons are integer multiples of his mass unit, having a nearly perfect correlation coefficient. For the mass of ψ(4160) in Ta- ble 3 we have used a recent measurement of Ablikim et al. (arXiv:0705.4500) which places m(ψ(4160)) at 4191.6 MeV/c2, which is 1.00175 · 31m(π0). In all there are 41 particles which follow the integer multiple rule. The data of Table 3 are plotted in Fig. 1, the line on Fig. 1 is determined by Eq.(2). 71", + "Electron flow in circular graphene quantum dots C. Schulz, R. L. Heinisch, and H. Fehske Institut f¨ur Physik, Ernst-Moritz-Arndt-Universit¨at Greifswald, 17487 Greifswald, Germany (Dated: March 12, 2018) Abstract We study the electron propagation in a circular electrostatically defined quantum dot in graphene. Solving the scattering problem for a plane Dirac electron wave we identify different scattering regimes depending on the radius and potential of the dot as well as the electron energy. Thereby, our particular focus lies on resonant scattering and the temporary electron trapping in the dot. 1 arXiv:1412.2539v1 [cond-mat.mes-hall] 8 Dec 2014 I. INTRODUCTION One of the most intriguing properties of graphene is the quasi-relativistic propagation of electrons. The intersection of energy bands at the edge of the Brillouin zone leads to a gapless conical energy spectrum in two inequivalent valleys. Low-energy quasiparticles are described by a massless Dirac equation1 and have a pseudospin which gives the contribution of the two sublattices of the graphene honeycomb lattice to their make-up. Chirality—the projection of the pseudospin on the direction of motion—is responsible for the conservation of pseudospin and the absence of backscattering for potentials diagonal in sublattice space.2 This particularity of the band structure allows perfect transmission for normal incidence at a n-p junction, termed Klein tunnelling.3,4 As Klein tunnelling prevents the electrostatic confinement of electrons, proposals for cir- cuitry have—in view of the linear energy dispersion—been modelled on optical analogues.5–7 Most notably n-p junctions could serve as lenses refocussing diverging electron rays.5 Inci- dentally, the experimental verification of Klein tunnelling was obtained in a Fabry-Perot interferometer composed of two n-p junctions.8 In this work we turn to the electron flow in a circular electrostatically defined quantum dot in graphene. For large dots—adequately described by ray optics—diffraction at the boundary results in two caustics inside the dot.9,10 For a small dot, however, wave optical features emerge. In this regime resonances in the conductance11,12 and the scattering cross section13,14 indicate quasi-bound states at the dot. Indeed, electrons can be confined in a circular dot surrounded by unbiased graphene as the classical electron dynamics in the dot is integrable and the corresponding Dirac equation is separable.11,15,16 In the following, we consider the scattering of a plane Dirac electron wave on a circular potential step. We study different scattering regimes, with a particular focus on quasi-bound states occurring for resonant scattering. II. THEORY For a dot potential that is smooth on the scale of the lattice constant but sharp on the scale of the de Broglie wavelength the low-energy electron dynamics in graphene is described 2 by the single-valley Dirac-Hamiltonian H = −i∇σ + V θ(R −r), (1) where R is the radius, V the applied bias of the gated region, and σ = (σx, σy) are Pauli matrices. We use units such that ℏ= 1 and the Fermi velocity vF = 1. In polar coordinates the Hamiltonian of unbiased graphene (V = 0) takes the form H =   0 e−iϕ \u0010 −i ∂ ∂r −1 r ∂ ∂ϕ \u0011 eiϕ \u0010 −i ∂ ∂r + 1 r ∂ ∂ϕ \u0011 0", + "ℏ= 1 and the Fermi velocity vF = 1. In polar coordinates the Hamiltonian of unbiased graphene (V = 0) takes the form H =   0 e−iϕ \u0010 −i ∂ ∂r −1 r ∂ ∂ϕ \u0011 eiϕ \u0010 −i ∂ ∂r + 1 r ∂ ∂ϕ \u0011 0  . (2) As a consequence of the spinor structure of the Hamiltonian the angular orbital mo- mentum L = x 1 i ∂ ∂y −y 1 i ∂ ∂x does not commute with the Hamiltonian, that is [L, H] = i \u0010 σx 1 i ∂ ∂y −σy 1 i ∂ ∂x \u0011 . However, the total angular momentum including orbital angular mo- mentum as well as pseudo-spin, J = L + σz/2, satisfies [J, H] = 0. To construct the eigenfunctions of the free Dirac equation in polar coordinates, Hψ(r, ϕ) = Eψ(r, ϕ), we need the eigenfunctions of J. The eigenfunctions of J to the eigenvalue m+1/2 are given by ξ+ m = eimϕ √ 2π  1 0  , ξ− m+1 = ei(m+1)ϕ √ 2π  0 1  . (3) In the next step we obtain the eigenfunction to J which is also an eigenfunction to H. For this, we make the ansatz ψm (r, ϕ) = f + m (r) ξ+ m (ϕ) + f − m+1 (r) ξ− m+1 (ϕ) . (4) Using the relation Hf ± m (r) ξ± m (ϕ) = \u0012 −i ∂ ∂rf ± m (r) ± im r f ± m (r) \u0013 ξ∓ m±1 , (5) we find the set of equations −i ∂ ∂rf + m (r) + im r f + m (r) = Ef − m+1 (r) , (6) −i ∂ ∂rf − m+1 (r) −im + 1 r f − m+1 (r) = Ef + m (r) , (7) which determine the radial wave functions f + m (r) and f − m+1 (r). Thereby, we have separated the Dirac equation in radial and angular parts. Substituting z = Er, f − m+1 (z) = Zm+1 (z) 3 \u0001 \u0001 \u0001i r t E V R FIG. 1. Sketch of the scattering geometry and the band structure for the scattering of a low- energetic electron at a circular dot in graphene. The dot is characterised by its radius R and the applied bias V . The incident plane wave with energy E > 0 (blue) corresponds to a state in the conduction band (upper cone). The reflected wave (purple) also lies in the conduction band while the transmitted wave inside the dot (red) occupies the valence band (lower cone). and if + m (z) = Zm (z), the above equations lead to the recurrence relations of the Bessel functions Zm. The eigenfunction of H in polar coordinates are thus given by ψm(r, ϕ) = −iZm (kr) 1 √ 2πeimϕ  1 0  + Zm+1 (kr) 1 √ 2πei(m+1)ϕ  0 1  , (8) where the wave number k = E. We now turn to the scattering of a plane wave on a", + "in polar coordinates are thus given by ψm(r, ϕ) = −iZm (kr) 1 √ 2πeimϕ  1 0  + Zm+1 (kr) 1 √ 2πei(m+1)ϕ  0 1  , (8) where the wave number k = E. We now turn to the scattering of a plane wave on a circular dot. The scattering geometry is illustrated in Fig. 1. The wave function of the incident electron, assumed to propagate in the x-direction, can be expanded according to ψi = 1 √ 2eikx  1 1  = 1 √ 2eikr cos ϕ  1 1  = 1 √ 2 ∞ X m=−∞ imJm (kr) eimϕ  1 1   (9) that is ψi = ∞ X m=−∞ √πim+1  −iJm (kr) 1 √ 2πeimϕ  1 0  + Jm+1 (kr) 1 √ 2πei(m+1)ϕ  0 1    . (10) This wave function is then matched with the reflected (scattered) and transmitted waves which are expanded in eigenfunctions in polar coordinates as well. The reflected wave is an outgoing wave. From the asymptotic behaviour of the Hankel function of the first kind H(1) m (kr) ∼ kr≫1 q 2 πkrei(kr−m π 2 −π 4) follows that for the reflected wave Zm(kr) = H(1) m (kr). 4 Introducing the scattering coefficients am, the reflected wave reads ψr = ∞ X m=−∞ √πim+1am  −iH(1) m (kr) 1 √ 2πeimϕ  1 0  + H(1) m+1 (kr) 1 √ 2πei(m+1)ϕ  0 1    . (11) The transmitted wave inside the dot has to be bounded at the origin. Only Bessel’s functions Jm satisfy this requirement. Thus the transmitted wave is given by ψt = ∞ X m=−∞ √πim+1bm  −iJm (qr) 1 √ 2πeimϕ  1 0  + αJm+1 (qr) 1 √ 2πei(m+1)ϕ  0 1    , (12) where the bm denote the transmission coefficients and α = sgn (E −V ) as well as q = α (E −V ). Requiring continuity of the wave functions at the boundary of the dot, that is ψi (r = R)+ ψr (r = R) = ψt (r = R), gives the conditions Jm (kR) + amH(1) m (kR) = bmJm (qR) , (13) Jm+1 (kR) + amH(1) m+1 (kR) = αbmJm+1 (qR) . (14) Solving these equations gives the scattering coefficients am = −Jm (Nρ) Jm+1 (ρ) + αJm (ρ) Jm+1 (Nρ) Jm (Nρ) H(1) m+1 (ρ) −αJm+1 (Nρ) H(1) m (ρ) , (15) bm = Jm (ρ) H(1) m+1 (ρ) −Jm+1 (ρ) H(1) m (ρ) Jm (Nρ) H(1) m+1 (ρ) −αJm+1 (Nρ) H(1) m (ρ) , (16) where we have introduced the size parameter ρ = kR and the modulus of the refractive index N = α (E −V ) /E. Using J−m = (−1)m Jm and H(1) −m = (−1)m H(1) m we find a−m = am−1 and b−m = αbm−1. The expressions for the reflected and transmitted wave can thus be rewritten as ψr = 1 √ 2 ∞ X m=0 im+1am  −iH(1) m (kr)  eimϕ e−imϕ", + "Using J−m = (−1)m Jm and H(1) −m = (−1)m H(1) m we find a−m = am−1 and b−m = αbm−1. The expressions for the reflected and transmitted wave can thus be rewritten as ψr = 1 √ 2 ∞ X m=0 im+1am  −iH(1) m (kr)  eimϕ e−imϕ  + H(1) m+1 (kr)  e−i(m+1)ϕ ei(m+1)ϕ    , (17) as well as ψt = 1 √ 2 ∞ X m=0 im+1bm  −iJm (qr)  eimϕ e−imϕ  + αJm+1 (qr)  e−i(m+1)ϕ ei(m+1)ϕ    . (18) 5 The electron density is given by n = ψ†ψ and the current by j = ψ†σψ where ψ = ψi +ψr outside and ψ = ψt inside the dot. The radial component of the current is given by jr = ψ† [σxcosϕ + σysinϕ] ψ = ψ†    0 1 1 0  cosϕ +  0 −i i 0  sinϕ  ψ. (19) Thus for the reflected wave the radial current takes the form jr r (ϕ) = 1 2 ∞ X m=0 (−i)m+1 a∗ m  iH(1)∗ m (kr)  e−imϕ eimϕ  + H(1)∗ m+1 (kr)  ei(m+1)ϕ e−i(m+1)ϕ     × ∞ X n=0 in+1an  −iH(1) n (kr)  e−i(n+1)ϕ ei(n+1)ϕ  + H(1) n+1 (kr)  einϕ e−inϕ    . (20) Simplifying this leads to jr r(ϕ) = ∞ X m,n=0 a∗ manin−m h\u0010 H(1)∗ m (kr) H(1) n (kr) + H(1)∗ m+1 (kr) H(1) n+1 (kr) \u0011 cos ((m + n + 1) ϕ) + i \u0010 H(1)∗ m (kr) H(1) n+1 (kr) −H(1)∗ m+1 (kr) H(1) n (kr) \u0011 cos ((m −n) ϕ) i . (21) In the far field jr r(ϕ) gives the angular scattering characteristic. In this limit (kr →∞) we can evaluate the above expression using the asymptotic behaviour of the Hankel functions. We obtain jr r (ϕ) = 4 πkr ∞ X m=0 |am|2 [cos ((2m + 1) ϕ) + 1] + 8 πkr X n0 the density is ring shaped. Panel A, B and D of Fig. 6 show the electron density for the modes a0, a1 and a3. Note that in panel D the small contribution of other modes of the dot to scattering leads to the slight asymmetry of the ring shaped electron density. Most importantly, the electron density inside the dot is dramatically increased which is a sign of temporary particle trapping at the scattering resonances. Finally, panel E of Fig. 6 shows the density profile for off-resonant scattering. In this case no electron trapping at the dot is observed but diffraction at the dot leads to an interference pattern in the passing wave. IV. SUMMARY In this work we have studied the scattering of a plane Dirac electron wave on a circular, electrostatically defined quantum dot in graphene. Due to the conical energy dispersion in graphene, electrons occupy non-evanescent states inside the dot even when their energy is below the dot potential. In particular, for a low energy of the incident electron, scattering resonances due to the excitation of normal modes of the dot appear. They are characterised by distinct preferred scattering directions. At the scattering resonances the electron density in the dot is strongly increased which indicates temporary electron trapping. ACKNOWLEDGEMENTS This work was", + "a low energy of the incident electron, scattering resonances due to the excitation of normal modes of the dot appear. They are characterised by distinct preferred scattering directions. At the scattering resonances the electron density in the dot is strongly increased which indicates temporary electron trapping. ACKNOWLEDGEMENTS This work was supported by the DFG Priority Programme 1459 Graphene. 1 P. A. M. Dirac, Proc. Roy. Soc. London, Ser. A 117, 612 (1928). 2 T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Jpn. 67, 2857 (1998). 11 3 O. Klein, Z. Phys. 58, 157 (1928). 4 M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nature Phys. 2, 620 (2006). 5 V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, Science 315, 1252 (2007). 6 C.-H. Park, Y.-W. Son, L. Yang, M. L. Cohen, and S. G. Louie, Nano Lett. 8, 2920 (2008). 7 J. R. Williams, T. Low, M. Lundstrom, and C. M. Marcus, Nature Nanotech. 6, 222 (2011). 8 A. Young and P. Kim, Nature Phys. 5, 222 (2009). 9 J. Cserti, A. P´alyi, and C. P´eterfalvi, Phys. Rev. Lett. 99, 246801 (2007). 10 M. M. Asmar and S. E. Ulloa, Phys. Rev. B 87, 075420 (2013). 11 J. H. Bardarson, M. Titov, and P. W. Brouwer, Phys. Rev. Lett. 102, 226803 (2009). 12 G. Pal, W. Apel, and L. Schweitzer, Phys. Rev. B 84, 075446 (2011). 13 R. L. Heinisch, F. X. Bronold, and H. Fehske, Phys. Rev. B 87, 155409 (2013). 14 A. Pieper, R. L. Heinisch, and H. Fehske, Europhys. Lett. 104, 47010 (2013). 15 P. Hewageegana and V. Apalkov, Phys. Rev. B 77, 245426 (2008). 16 P. Hewageegana and V. Apalkov, Phys. Rev. B 79, 115418 (2009). 12", + "arXiv:0905.4765v3 [physics.plasm-ph] 16 Sep 2009 Extension of the electron dissipation region in collisionless Hall MHD reconnection Brian P. Sullivan,∗A. Bhattacharjee,† and Yi-Min Huang‡ Center for Integrated Computation and Analysis of Reconnection and Turbulence Institute for the Study of Earth, Oceans, and Space University of New Hampshire, Durham, NH 03824, USA (Dated: today) This paper presents Sweet-Parker type scaling arguments in the context of hyper-resistive Hall magnetohyrdo- dynamics (MHD). Numerical experiments suggest that both cusp-like and modestly more extended geometries are realizable. However, the length of the electron dissipation region, which is taken as a parameter by several recent studies, is found to depend explicitly on the level of hyper-resistivity. Furthermore, although hyper- resistivity can produce more extended electron dissipation regions, the length of the region remains smaller than one ion skin depth for the largest values of hyper-resistivity considered here–significantly shorter than current sheets seen in many recent kinetic studies. The length of the electron dissipation region is found to depend on electron inertia as well, scaling like (me/mi)3/8. However, the thickness of the region appears to scale similarly, so that the aspect ratio is at most very weakly dependent on (me/mi). I. INTRODUCTION The problem of fast magnetic reconnection in collisionless or weakly collisional plasmas has become a subject of great interest in recent years, in large part due to its relevance to impulsive phenomena such as magnetospheric substorms, so- lar and stellar flares, and the sawtooth crash in tokamaks. A key development has been the realization that non-ideal terms in the generalized Ohm’s law (including electron inertia, elec- tron pressure gradient, and Hall terms) provide a pathway to the onset of fast reconnection. This subject has a history span- ning nearly two decades, with important contributions from both the fusion and space plasma communities (cf. [1, 2] and references therein). It has been suggested that the Hall term in particular plays a key role in localizing the electron dif- fusion region. Within the framework of Hall MHD (or two- fluid) studies, there has been consensus on this point, though the issue of reconnection scaling has remained controversial. Some studies of reconnection in Hall MHD systems have re- ported a “universal” normalized reconnection rate of ∼0.1, independent of dissipation mechanism [3, 4, 5, 6], or system size[7, 8, 9]. However, other studies[10, 11, 12, 13, 14, 15, 16] have found a broad range of dependencies on plasma param- eters such as the ion skin and electron skin depths, and on boundary conditions, appearing to refute the claim of univer- sality. In an interesting sequence of studies based on fully kinetic simulations employing particle-in-cell (PIC) methods [17, 18, 19], questions have been raised regarding the conclusion, ob- tained from Hall MHD studies, that the electron diffusion re- gion is highly localized. In these PIC studies the electron dif- fusion region is not so localized. Rather it is elongated in the outflow direction to lengths on the order of 10di ( where di = c/ωpi is the ion skin depth) for electron to ion mass ratios of the order of 1/100, when Hall", + "these PIC studies the electron dif- fusion region is not so localized. Rather it is elongated in the outflow direction to lengths on the order of 10di ( where di = c/ωpi is the ion skin depth) for electron to ion mass ratios of the order of 1/100, when Hall MHD simulations ∗Electronic address: brian.sullivan@unh.edu †Electronic address: amitava.bhattacharjee@unh.edu ‡Electronic address: yimin.huang@unh.edu produce localized (≤di length) electron current sheets (see e.g. [20, 21]). This raises the following interesting question: are fully kinetic simulations necessary to produce elongated electron current sheets? More specifically, is it possible to re- alize elongated electron current sheets within the framework of Hall or extended MHD models by means of a generalized Ohm’s law that includes additional closure terms and param- eterizes physical processes that may have kinetic origin? If the answer to this question is yes, it may facilitate greatly the capability of global multi-fluid models to represent the effects of kinetic physics at small scales. A recent study by Chac´on et al. [22] presented scaling argu- ments and simulation results which indicated that in 2D elec- tron MHD (EMHD) systems with electron viscosity (hyper- resistivity), the electron dissipation region can take on a wide range of geometries; the governing equations permit the as- pect ratio (width/length) of the electron dissipation region to be of order unity (cusp like) or much smaller, i.e., leading to an extended electron current sheet. In the present work, we re-derive Chac´on’s EMHD scaling in perhaps a more trans- parent way, via Sweet-Parker type arguments about quantities at the upstream and downstream edges of the electron dissipa- tion region. Similar approaches have been employed recently by Malyshkin [23], and also by Uzdensky [24]. These scaling results are then benchmarked numerically by two-fluid simu- lations of island coalescence both with and without electron inertia. We emphasize that neither this study, nor any other to date has produced a general method for analytically predict- ing the length of the dissipation region, which is perhaps the most important factor in determining whether the reconnec- tion in a given system is fast or slow. We show, in particular, that the length of the electron diffusion region cannot be sim- ply assumed to be a parameter fixed only by external bound- ary conditions, but depends on the mechanism that breaks the frozen-in condition (electron viscosity in this case). The organization of this paper is as follows. In section II we describe the simulation model and equilibrium profiles. In section III, we derive our scaling arguments. In section IV we describe the diagnostics used in analyzing simulation results. In section V we present and discuss the results of the simulations included in this study. The main conclusions are 2 FIG. 1: (color online) Initial out-of-plane current, Jz(x, y) (orange scale), and magnetic field lines (black contours). summarized in section VI. II. SIMULATION MODEL A. Model Equations Our simulations are based on the two-fluid equations in- cluding hyper-resistivity. These equations in normalized form are [8]: n (∂tVi + Vi · ∇Vi) = J × B −∇p, (1) ∂tB′ = −∇×", + "scale), and magnetic field lines (black contours). summarized in section VI. II. SIMULATION MODEL A. Model Equations Our simulations are based on the two-fluid equations in- cluding hyper-resistivity. These equations in normalized form are [8]: n (∂tVi + Vi · ∇Vi) = J × B −∇p, (1) ∂tB′ = −∇× E′, (2) E′ = −Vi × B + 1 n (J × B′ −∇pe) −ηH∇2J, (3) ∂tn + Vi · ∇n = −n∇· Vi, (4) where pe = nTe , pi = nTi , p = pi + pe, B′ = \u00001 −d2 e∇2\u0001 B, and Ve = Vi −J/n , J = ∇× B. For simplicity we assume an isothermal equation of state for both electrons and ions (qualitatively similar to the adi- abatic case), and thus take Te and Ti to be constant. The normalizations of Eqns. (1)-(4) are based on constant refer- ence values of the density n0 and the reconnecting compo- nent of the magnetic field Bx0, and are given by (normalized →physical units): t →ωcit, ωci = eB0/(mic), x →x/di, where di,e = c/ωpi,e, and ω2 pi,e = 4πn0e2/mi,e, n →n/n0, B →B/B0, Vi,e →Vi,e/VA, where VA = ωcidi = B0/(4πn0mi)1/2, Ti,e →Ti,e4πn0/B2 0, pi,e →pi,e4π/B2 0, and J →J/(n0eVA). Our algorithm employs fourth-order accurate spatial finite differencing and the time stepping scheme is a second-order accurate trapezoidal leapfrog [25, 26] . We consider a square, 2D slab with physical dimensions L × L = 12.8di × 12.8di (so that L = 12.8 in normalized units) and periodic bound- ary conditions imposed at x = ±L/2, and y = ±L/2. The simulation grid is nx × ny = 512 × 512, yielding grid scales of ∆x = ∆y = 0.025. Note that compared to the simple form of Ohm’s Law in resistive MHD, the generalized Ohm’s Law in this two-fluid system (i.e. Eq. (3) above), contains four additional non-ideal terms: the Hall term (∝J × B), the electron pressure gradient term (∇pe), the electron iner- tia terms, (which are manifested by the d2 e terms in the def- inition of B′), and the hyper-resistive term (∝∇2J). Note that of these non-ideal terms, only the hyper-resistive term is dissipative. Although hyper-resistivity (also known as elec- tron viscosity) is generally negligibly small in collisional plas- mas, it can be significant in a collisionless plasma. For a more thorough discussion of hyper-resistivity see, for exam- ple, Refs. [27, 28, 29, 30, 31, 32, 33, 34] . We consider the case of zero resistivity, and the model described here is thus intended to apply to collisionless plasmas. B. Initial Equilibrium We begin with a two-dimensional (2D) equilibrium simi- lar to that used by Longcope and Strauss (1994), shown in Fig (1). The normalized magnetic field and density profiles in our initial equilibrium are given by: Bx(x, y) = B0 \u0014 cos \u00122πy L \u0013 sin \u00122πx L \u0013\u0015 ˆx, (5) By(x, y) = −B0 \u0014 cos \u00122πx L \u0013 sin \u00122πy L \u0013\u0015 ˆy, (6) n(x, y) = 1 + 1 −B2 2(Ti + Te). (7) The boundary conditions are periodic", + "equilibrium are given by: Bx(x, y) = B0 \u0014 cos \u00122πy L \u0013 sin \u00122πx L \u0013\u0015 ˆx, (5) By(x, y) = −B0 \u0014 cos \u00122πx L \u0013 sin \u00122πy L \u0013\u0015 ˆy, (6) n(x, y) = 1 + 1 −B2 2(Ti + Te). (7) The boundary conditions are periodic in both directions. As required by these boundary conditions, B is periodic under y →y + L and x →x + L . Note that the normalization parameter B0 is equal to 2π/L = 2π/12.8 = 0.4908. The density profile is chosen to satisfy the total pressure balance condition, which in normalized form is given by n(Ti + Te) + 1 2B2 = constant. (8) Unless otherwise stated, we take the (constant) total temper- ature to be Ttot = Ti + Te = 1.0, or in physical units 4πn0Ttot/B2 x0 = 1, so that the plasma β has a (minimum) value of 11.45 where Bx = 0.4908 and n = 1.3795. The density reaches a maximum value of n = 1.5 in the center of the simulation domain (x = 0, y = 0), where B = 0. Initially the current is carried entirely by the ions. To prevent energy buildup at the grid scale, the simulations include fourth order dissipation in the density and momentum equations of the form µ4∇4 where µ4 = 5.1 · 10−7. To avoid physically 3 artificial effects that can arise from exact reflection symme- tries of the initial condition, a small amount of random noise is added to the magnetic field and ion current at the levels | ˜Bmax| ≈10−5, | ˜Jimax| ≈10−5. This initial equilibrium configuration contains four mag- netic islands. Diagonally adjacent islands contain (out-of- plane) currents of like sign and are therefore mutually at- tracted. The islands with currents of opposite sign are of course mutually repelled. At rest these attractions and re- pulsions balance so that the equilibrium is metastable: ei- ther the upper right and lower left islands can coalesce at the center of the simulation domain, or the upper left and lower right islands can do so. Due to the symmetry of this sys- tem, whichever pair coalesces at the center, the opposite pair of islands will be repelled from the center coalescing at one of the corners of the simulation box. The symmetry of this metastable state is broken by giving the system an initial in- compressible in-plane velocity perturbation of the form: Vi(x, y) = V0 \u0014 sin \u00122πy L \u0013 ˆx + sin \u00122πx L \u0013 ˆy \u0015 . (9) Unless otherwise stated, the magnitude of this initial velocity is V0 = 0.1 in normalized units. III. SCALING ARGUMENTS In systems where the Hall term is important, the non-ideal region takes on a two-scale structure: an outer region where the ions are demagnetized but the electrons are still frozen in, and an inner region where the electrons are demagnetized as well. It should be emphasized that in the outer part of the non- ideal region, although the ions are demagnetized, there is no dissipation occurring there. Here", + "region where the ions are demagnetized but the electrons are still frozen in, and an inner region where the electrons are demagnetized as well. It should be emphasized that in the outer part of the non- ideal region, although the ions are demagnetized, there is no dissipation occurring there. Here we will refer to this outer, non-dissipative zone as the “Hall region.” The inner, dissipa- tive region, where neither species is magnetized, and where (in our case) electron viscosity is the dominant term, will be referred to as the “electron diffusion region.” Let δe, and Le denote the thickness and length of the electron diffusion re- gion, respectively. In the following scaling arguments we neglect the effects of electron inertia, which is valid when the smallest scale of interest is much larger than the electron inertial length, de. We also suppose that the spatial scale of the electron dissi- pation region is small sufficiently small compared to the ion inertial length, di, so that ion motion may be neglected. In other words, we consider an EMHD model. In the regime where the above conditions are met, the dominant terms on the right hand side of the generalized Ohm’s Law (Eq. (3)) are the Hall (proportional to J × B) term, and the hyper-resistive term (proportional to ηH∇2J). Let us consider the how these two terms conspire to influence the geometry of the electron region. Note that the arguments made here are very similar to those made by Uzdensky in a recently submitted paper [24]. From this point on we will work in a co-ordinate system rotated 45 degrees clockwise (ˆx →(1/ √ 2)(ˆx −ˆy)) with re- spect to the system coordinates described in Section (II B), so that we call the inflow direction “y” and the outflow direction “x”. In the case under consideration, there are two forces act- ing on the electrons as they flow away from the x-point: the J × B force accelerates the electrons away from the x-point along the x and z directions in a channel of width ∼δe, while the hyper-resistive (electron viscosity) term resists the shear in the electron fluid caused by the localized outflow. The peak electron outflow in steady state occurs at the point where these two opposing forces balance. This balancing of terms at the point of peak outflow is found to be well satisfied in our simu- lation results. Here we will take the half length (Le/2) of the layer to be the distance from the x-point/stagnation point to the location of peak electron outflow. Let Bey, Vex, and Jez denote the magnetic field, electron outflow velocity, and out- of-plane current density at the point (x, y) = (Le, 0). Then we have Ex(Le/2, 0) = BeyJez cne −ηH∇2Jex = 0, =⇒BeyJez cne ≃ηH neVex δ2e , =⇒Vex ≃BeyJezδ2 e cn2e2ηH , (10) where the approximation that ∇2 ∼δ−2 e holds provided that δe ≪Le. Between the inner and outer region, the electrons move at the E × B drift speed. So, at the downstream edge of the electron region, Bey", + "=⇒BeyJez cne ≃ηH neVex δ2e , =⇒Vex ≃BeyJezδ2 e cn2e2ηH , (10) where the approximation that ∇2 ∼δ−2 e holds provided that δe ≪Le. Between the inner and outer region, the electrons move at the E × B drift speed. So, at the downstream edge of the electron region, Bey = cEz/Vex. In steady state, ∂tB = ∇× E = 0 implies that Ez is spatially uniform. Thus, Ez(Le, 0) = Ez(0, 0) = ηH∇2Jz(0, 0) ≃ηHBex/δ3 e, (where Bex is the magnitude of the reconnecting component of the magnetic field at (x, y) = (0, δe)) yielding: Bey ≃cηHJz0 Vexδ2e . (11) Substituting this value for Bey into Eq. (10) gives [24, 35]: |Vex| = |Vez| = Jz ne ∼1 ne cBex 4πδe = diVA δe = deVAe δe , (12) where VA = Bex/√4πmin, VAe = Bex/√4πmen, di = c/ωpi, and de = c/ωpe. We must emphasize here the quan- tity (diVA) = (deVAe) = (cBex/4πne) is independent of the mass of either species. The present scaling arguments are essentially EMHD arguments, and as such there is no finite di. In the absence of electron inertia, there is also no de in an EMHD model. Uniformity of Ez holds across the electron current sheet as well, requiring |Vey|Bex = ηH∇2Jz0 ≃ηH Bex δ3e . (13) Finally, taking the plasma to be incompressible (rigorously valid in the EMHD limit), requires that |Vey|Le ≃|Vex|δe, we can solve for δe, |Vex|, |Vey|, and Ez in terms of VA, di, ηH, 4 and Le: δe = \u0012ηHLe diVA \u00131/3 , (14) |Vey| = (diVA) Le , (15) |Vex| = |Vez| = (diVA) δe = (diVA)4/3 (ηHLe)1/3 , (16) cEz = (diVA)Bex Le = 1 Le \u0012 cB2 ex 4πne \u0013 (17) Equation (14) predicts exactly the same scaling as that found by Chac´on et al. (compare to Eq. (8) of reference [22]). How- ever, as in that study, the length of the electron dissipation region remains as a free parameter. This scaling permits (as stated by Chac´on) both cases in which δe ∼Le ∼η1/2 H , or δe ∼(ηH/VA)1/3 ≪Le. However, this statement may lead one to believe that there exists a bifurcated solution, which permits only the two aforementioned scalings. In fact, neither the arguments presented here nor those made in Refs. [22] or [23] places any limit on the scaling of Le, thus failing to an- swer the question of whether the reconnection in any given system will be fast or slow. So in fact, a continuum of scal- ings may exist. The reconnection rate given in Eq. (17) is not explicitly dependent on ηH, but may implicitly depend on ηH, depending on how Le scales. Several limitations of the above scaling arguments are worth noting. First, the theory has as- sumed that ion motion is negligible on the spatial and tempo- ral scale of the reconnection process. Second, the effects of electron inertia have been neglected. Third, electron pressure gradients have been assumed to be negligible, largely for the sake of simplicity; on scales >∼di, electron pressure gradients are generally expected", + "that ion motion is negligible on the spatial and tempo- ral scale of the reconnection process. Second, the effects of electron inertia have been neglected. Third, electron pressure gradients have been assumed to be negligible, largely for the sake of simplicity; on scales >∼di, electron pressure gradients are generally expected to be of the same order as the J × B term, as can be seen from Eq. (3). The first two of these points limit the applicability of these arguments to scales ℓthat sat- isfy the condition de ≪ℓ≪di. A further, more subtle point, which was mentioned above, is that the downstream force bal- ance criterion may not hold in all cases. However, even if there is no downstream force balance, the same scalings can be derived by enforcing steady state of the 2D EMHD equa- tion. In two dimensions, the magnetic field can be expressed without loss of generality as B = ˆz×∇ψ+Bz(x, y)ˆz. In this case, the in plane and out-of-plane components of the EMHD equation can be expressed (in normalized form) as: ( ∂tψ = −B⊥· ∇⊥Bz −ηH∇2 ⊥Jz = −Ez = const., ∂tBz = −B⊥· ∇⊥Jz + ηH(∇2 ⊥)2Bz = 0. (18) In steady state, again taking ∂x ≈L−1 e , ∂y ≈δ−1 e , ∇2 ⊥≈ δ−2 e , these two equations become: ( BexBez Le ∼BeyBez δe ∼ηH Bex δ3e , B2 ex δeLe ∼ηH Bez δ4e , (19) and eliminating Bez from these two equations, again yields: δe = (ηHLe/Bx)1/3, the same scaling given by Eq. (14), and FIG. 2: (color online) Dissipation region in a simulation with me = 0. top panel: 1D cuts of electron and ion outflow veloci- ties, Vex, & Vix. The vertical lines mark the location of peak elec- tron outflow (ends of electron diffusion region) and the points where electron and ion velocities come together. second panel: Jz(x, y); dashed lines mark edges of electron diffusion region. third panel: Jiz(x, y); dashed lines mark the locations where the ion and electron flows come together. fourth panel: ion outflow,Vix(x, y), bottom panel: electron outflow, Vex(x, y) equivalent scalings come about for the other quantities as well. This approach is perhaps more general, as it doesn’t require force balance at the downstream edge, though it may be less physically transparent. A general theory which predicts the length of the dissipation region in collisionless Hall MHD is in many ways the ultimate pursuit of all theoretical work on reconnection today, and as of yet no theory has delivered that result, except in special cases (e. g. [11, 19]). Lacking such a general theory, we now to numerical experiments to explore the dependence of Le on ηH. 5 IV. DESCRIPTION OF DIAGNOSTICS Before describing the results of our numerical experiments, we must describe the diagnostics employed in quantifying the extent of the non-ideal regions for each species. Near the cen- ter of the current sheet, the out-of-plane current density, Jz, is due primarily to the electrons. In this study the width of the electron dissipation region is taken to be the full", + "describe the diagnostics employed in quantifying the extent of the non-ideal regions for each species. Near the cen- ter of the current sheet, the out-of-plane current density, Jz, is due primarily to the electrons. In this study the width of the electron dissipation region is taken to be the full width at half max of the out-of-plane current, Jz. The downstream edge of the electron dissipation region is taken to be the lo- cation of the peak in the electron outflow speed. Outside of the non-ideal region, the two species flow roughly together at the E × B drift velocity. Both the upstream and downstream edges of the non-ideal dissipation region are taken to be the first point outside the electron dissipation region at which the electron and ion velocities differ by less than, say, ten per- cent. For example, to determine the location of the down- stream edge, we begin at the downstream edge of the elec- tron dissipation region and search along x until the quantity \u0010 1 −vi(x,0) ve(x,0) \u0011 ≤0.1. Figure (2) shows the shape of the ion and electron dissipation regions at late time in a simulation with me = 0. The top panel shows 1D cuts of Vex (solid), and Vix (dotted) along the outflow (x) direction. The inner pair of dashed vertical lines in this panel indicate the locations of peak electron outflow, which are taken to be the ends of the electron dissipation region. The outer pair of vertical lines mark the point where \u0010 1 −vi(x,0) ve(x,0) \u0011 = 0.1, i.e. the ends of the Hall region. The second panel from the top shows the out of plane current, Jz(x, y), along with dashed lines indicating the extent of the electron dissipation region. The third panel shows Jiz, the out-of-plane ion current, with the dashed lines now indicating the extent of the Hall region in the x-y plane. The lower two panels show Vix, and Vex in the x-y plane. For comparison, corresponding results from a simulation with me/mi = 1/25 are shown in Fig. (3). The point of peak electron outflow is further downstream than that observed in the simulation with no electron mass. This extension of the current sheet was seen in an earlier study [37], but was not well explored. Note that the electron outflow jets extend well beyond the boundaries of the electron diffusion region, and into the boundaries of the non-dissipative, Hall region. This feature is consistent with results PIC simulations. A recent paper by Hesse et al. [36] provides a detailed discussion of the non-dissipative nature of these outflow jets in the context of a kinetic system. The shape and extent of the Hall region are basically unchanged by varying ηH and me/mi. So we will focus here on the electron dissipation region. In examin- ing the scaling geometry of the electron dissipation region, we will focus on quantities defined at the centers of the upstream and downstream edges. The upstream field, Bex, upstream density, ne, and electron inflow speed Vey, are measured at the point (x,", + "here on the electron dissipation region. In examin- ing the scaling geometry of the electron dissipation region, we will focus on quantities defined at the centers of the upstream and downstream edges. The upstream field, Bex, upstream density, ne, and electron inflow speed Vey, are measured at the point (x, y) = (0, δe), while the electron outflow speed is measured at the point (x, y) = (Le, 0), which, as mentioned above is taken to be the location of the peak electron outflow. FIG. 3: (color online) Dissipation region in a simulation with me/mi = 1/25. top panel: 1D cuts of electron and ion outflow velocities, Vex, & Vix. The vertical lines mark the location of peak electron outflow (ends of electron diffusion region) and the points where electron and ion velocities come together. second panel: Jz(x, y); dashed lines mark edges of electron diffusion region. third panel: Jiz(x, y); dashed lines mark the locations where the ion and electron flows come together. fourth panel: ion outflow,Vix(x, y), bottom panel: electron outflow, Vex(x, y) V. NUMERICAL RESULTS A. Hyperresistive Scaling Because, as mentioned above, no ab initio theoretical for- mulation of reconnection has yet successfully predicted the length of the current sheet in a Hall MHD system, we here begin our numerical investigation by describing the observed dependence of Le on varying hyper-resistivity, ηH. Figure (4) presents measurements of the length of the electron dissipa- 6 FIG. 4: (color online) Time series of the length of the electron diffu- sion region, Le, for varying levels of ηH (upper panel), and scaling of median plateau values of Le vs. ηH (lower panel). tion region from five two-fluid simulations of magnetic is- land coalescence. Time series of Le are shown in the top panel of the figure. The five, solid curves represent the results of simulations with varying levels of hyper-resistivity, which do not include the effects of electron inertia. Notice that by tΩci ∼6.5 each of these simulations has achieved a relatively stable value of Le. The lower panel of the figure plots the median value of Le from the plateau phase (tΩci ≥6.5) ver- sus ηH. The scaling of Le is quite clear, as shown by the dashed line (of slope η1/3 H ) in the lower panel. Empirically, Le ∝η1/3 H in this modestly sized, periodic, coalescing sys- tem. How does this empirical result for Le relate to the scal- ings predicted in the previous section? If one simply inserts Le ∝η1/3 H into Eq. (14), one finds δe ∝η4/9 H . Time series of δe are plotted in the upper panel of Fig. (5). As was the case with Le, the thickness of the electron dissipation region, reaches a stable value after tΩci ∼6.5. In the lower panel of Fig. (5), the median plateau value of δe is plotted versus ηH. The dotted line, plotted for reference, represents a scaling of η4/9 H , in reasonable agreement with the data points. In the case of the lowest value of ηH presented in this figure, the current sheet thickness drops", + "Fig. (5), the median plateau value of δe is plotted versus ηH. The dotted line, plotted for reference, represents a scaling of η4/9 H , in reasonable agreement with the data points. In the case of the lowest value of ηH presented in this figure, the current sheet thickness drops nearly to the grid scale. So that point may not be entirely reliable. As a comparison to the results of Chac´on et al., in Fig. (6) we plot δe(Bex/Le)1/3, analogous to Chac´on’s Fig. 3. The upper dotted line represents η1/3 H , though the slope is actually closer to η4/9 H (as shown by the lower dotted line). Note that if Le ∼η1/3 H (purely an empir- ical result), and δe ∼η4/9 H (in accordance with the predicted scaling of δe, and the observed scaling of Le), then the aspect ratio of the dissipation region has an extremely weak depen- FIG. 5: (color online) Time series of the thickness of the electron dif- fusion region, δe, for varying levels of ηH (upper panel), and scaling of median plateau values of δe vs. ηH (lower panel). FIG. 6: Scaling of δe(Bex/Le)1/3 vs. ηH dence on hyper-resistivity δe/Le ∼η1/9 H . Figure (7) presents the scaling of the aspect ratio of the electron dissipation re- gion. Time series of δe/Le are plotted in the upper panel of the figure. In the lower panel of the figure, diamonds repre- sent the median plateau value of the aspect ratio. As expected , the aspect ratio varies extremely little across the range of hyper resistivities considered here. These scalings of δe and Le have been further tested in analogous systems of twice the physical size (Lx = Ly = 25.6) and half the physical size 7 FIG. 7: (color online)Time series of the aspect ratio of the electron diffusion region, δe/Le, for varying levels of ηH (upper panel), and scaling of median plateau values of δe/Le vs. ηH (lower panel). (Lx = Ly = 6.4) of the system described above. The results are shown in Fig. (8). Not only the scaling of Le, but even its absolute numerical value is practically unchanged by dou- bling the system size, as shown by the triangles and diamonds in the upper panel of Fig. (8). The scaling of Le does how- ever break down if the system size is too small, in which case the small size of the coalescing flux bundles limits the poten- tial extent of Le. The dependence of δe on ηH, is unchanged by doubling or halving the system size. Thus for sufficiently large systems, the scalings found here do not appear to depend on the system size. This behavior appears to be quite differ- ent from that found in Ref. [16], which used a different initial condition and did not achieve steady state at any point during the evolution of the system. Next we turn our attention to the scaling of the reconnection rate. Time series of the electric field measurements are noisy. So, we consider measurements averaged over a window in time: ER ≡", + "initial condition and did not achieve steady state at any point during the evolution of the system. Next we turn our attention to the scaling of the reconnection rate. Time series of the electric field measurements are noisy. So, we consider measurements averaged over a window in time: ER ≡ √ne B2ex ∂Az ∂t (20) where Az is the component of the magnetic vector poten- tial in the ignorable direction (i.e. the flux function up to a sign), and ⟨⟩denotes an average over an interval of one ion cyclotron time. Here the reconnection has been normalized by the the local upstream values of the magnetic field and density, Bex, and ne, which as mentioned earlier are mea- sured at the point upstream where the out of plane current has fallen to half of its maximum value. Figure (9) shows normalized reconnection rates computed from simulation re- sults. Again the upper panel shows time series, while the lower panel shows median plateau values. These simulations achieve a stable plateau value for ⟨ER⟩between tΩci = 6.5 and tΩci = 8.5. In the lower panel of Fig. (9) the diamonds represent the plateau value of normalized reconnection rate from the time series shown in the upper panel. The stars rep- resent plateau values of the reconnection rate over the same range of hyper-resistivities in the larger (Lx = Ly = 25.6di) system. The dotted line, plotted for reference represents a slope of η−1/3 H , as predicted by Eq. (17) for our observed scal- ing of Le ∝η1/3 H . The observed reconnection rates appear to be within a factor of two of the predicted scaling. We note that although it is common practice to normalize the reconnection rate using local values of the magnetic field and density, these values are not necessarily values that fairly characterize the system. In this system for example, the meaningful physical purpose of the normalized reconnection rate is to character- ize the time scale of the process of reconnecting a significant amount of the flux that was contained in the islands at t = 0. To this end, it makes sense to use typical “global” values of magnetic field and density, which characterize the initial con- dition. Figure (10) shows reconnection rates from the same simulations as in Fig. (9). However, in Fig. (10) the reconnec- tion rates have been normalized to the global peak initial val- ues of B and n. Again the top panel shows time series, while the lower panel illustrates the scaling of the plateau value. These “globally” normalized reconnection rates appear to be nearly constant with a value of approximately 0.1−0.2 across the range of hyper-resistivities considered here. Note that the “raw,” unnormalized reconnection rates are also nearly inde- pendent of ηH, while the upstream field and density do appear to depend on the dissipation coefficient. Figure (10) makes clear that although the locally normalized reconnection rate depends on ηH, the time needed to reconnected a significant fraction of the initial flux in this system (as reflected by the “globally” normalized reconnection rate) is", + "upstream field and density do appear to depend on the dissipation coefficient. Figure (10) makes clear that although the locally normalized reconnection rate depends on ηH, the time needed to reconnected a significant fraction of the initial flux in this system (as reflected by the “globally” normalized reconnection rate) is roughly indepen- dent of the dissipation region, and in this sense, the reconnec- tion observed here is fast. B. Electron Mass Scaling We have observed above that the dimensions of the elec- tron current sheet can be extended in cases which include electron inertia, as compared to hyper-resistive simulations that do not include the effects of finite electron mass (see Figs. 4 and 5). Additionally, for a given mass ratio, there ap- pears to be a threshold value of ηH below which Le is deter- mined by the mass ratio, and the effects of ηH become neg- ligible. The present study will not treat the general scaling of systems in which the effects of hyper-resistivity and elec- tron inertia are comparable. However, we briefly present scal- ing results from simulations with varying electron mass, for η = 0, ηH = 1.0 × 10−5. The top panel of Fig. 11 shows time series of Le for mass ratios varying from me/mi = 1 25– 1 200, and again the bottom panel plots the plateau values of Le vs. me/mi. The dotted line in the lower panel represents an estimated scaling of Le ∝(me/mi)3/8. This scaling is consistent with that found by Shay et al. [19], in the context of PIC simulations. The X in the figure denotes an extrapo- 8 FIG. 8: (color online)Scaling of the Length of the electron diffusion region, Le vs. hyperresistivity, ηH for varying system size(upper panel), and scaling of the electron diffusion region thickness, δe (lower panel). lated value of Le ≈15de (or about 1/3 of an ion skin depth) for a physical electron/ion mass ratio of 1/1836. Note that Shay et al. extrapolate to a value of Le ≃0.6di for a phys- ical mass ratio, less than a factor of two different from our two-fluid result. This scaling can be understood using argu- ments presented in the aforementioned study by Shay et al.; the method is similar to that used earlier in this paper to de- rive hyper-resistive scalings. Suppose that the dominant terms in the electron momentum equation at the downstream edge of the electron dissipation region are the electron inertia term and the Hall (J × B) term. Then in steady state (∂t = 0), and assuming the current to be entirely carried by the electrons at the scale of the dissipation region (the EMHD approxima- tion) we have the following steady state electron momentum equation: nmeVe · ∇Ve = ne(Ve × B), (21) FIG. 9: (color online)Time series of the locally normalized reconnec- tion rate, ⟨ER⟩L, for varying levels of ηH (upper panel), and scaling of median plateau values of ⟨ER⟩L vs. ηH (lower panel) FIG. 10: (color online)Time series of the globally normalized re- connection rate, ⟨ER⟩G, for varying levels of ηH (upper", + "9: (color online)Time series of the locally normalized reconnec- tion rate, ⟨ER⟩L, for varying levels of ηH (upper panel), and scaling of median plateau values of ⟨ER⟩L vs. ηH (lower panel) FIG. 10: (color online)Time series of the globally normalized re- connection rate, ⟨ER⟩G, for varying levels of ηH (upper panel), and scaling of median plateau values of ⟨ER⟩G vs. ηH (lower panel) whose x component can be expressed as: ∂ ∂x \u0012 me V 2 ex 2 \u0013 = eVezBy. (22) This equation can be integrated from the x-point to the down- stream edge to arrive at an equation for Le. However, in order to integrate the above equation, we need to know something 9 FIG. 11: (color online)Time series of the length of the electron dif- fusion region, Le, for varying values of me/mi (upper panel), and scaling of median plateau values of Le vs. me/mi (lower panel). about the x dependence of By. In agreement with the PIC re- sults of Shay et al. [19] we have found that in our two-fluid model the profile of By along the outflow direction in the dis- sipation region does not appear to depend noticeably on the mass ratio, as can be seen in Fig. (12). Taking Vez ∼VAe, assuming By to vary linearly with distance from the x-point as By(x) = B′ yx, and integrating the above equation yields: Le ∼ \u0012me mi \u00133/8 \u0012 Ez B0VA \u00131/2 \u0012 B0 diB′y \u00133/4 di (23) This expression differs slightly from that obtained by Shay et al. They give the exponent of the (B0/diB′ y) factor as 1, whereas we find it to be 3/4. Note that due to this difference our expression is entirely independent of mi (as it should be since there was no mi in Eq. (21), whereas theirs contains an implicit mi dependence (via the VA in the second factor). This difference is not likely to become apparent in numerical experiments, as neither B0 nor B′ y appear to depend on the mass ratio. Biskamp et al. [39] have made different scaling arguments using the observed uniformity of the canonical mo- mentum (F = jz −d2 eψ) along the upstream edge of the diffu- sion region, which yield a scaling of Le ∼(me/mi)1/3. This scaling could also agree reasonably well with the results of our numerical experiment. Resolving the difference between a scaling of 3/8 and 1/3 is difficult with only one decade of mass ratios. However, it is encouraging that our two-fluid scalings reasonably reproduce the mass ratio scaling seen in PIC simulations. The scaling of the dissipation region thickness, δe, with me/mi is shown in Fig. 13. The dotted line in the lower panel indicates a scaling of (me/mi)1/3. This is somewhat stronger than the scaling [δe/di ∼(me/mi)1/4] observed by Daugton FIG. 12: (color online)Profiles of By(x) along the outflow direction at t = 11.0 in simulations with varying mass ratios. Note that the slope of By appears to be nearly independent of me/mi at x=0. FIG. 13: (color online)Time series of the thickness of the electron", + "∼(me/mi)1/4] observed by Daugton FIG. 12: (color online)Profiles of By(x) along the outflow direction at t = 11.0 in simulations with varying mass ratios. Note that the slope of By appears to be nearly independent of me/mi at x=0. FIG. 13: (color online)Time series of the thickness of the electron diffusion region, δe, for varying values of me/mi (upper panel), and scaling of median plateau values of δe vs. me/mi (lower panel). et al. [17], while showing good agreement with that observed by Dorfman et al. [δe/di ∼(me/mi)1/3] in kinetic simula- tions of reconnection in MRX [38] (although in those simu- lations the frozen-in condition is broken by electron pressure agyrotropy rather than hyper-resistivity). Here the extrapo- lated valued of δe for a physical mass ratio is approximately 1de. The aspect ratio of the electron diffusion region does not appear to depend significantly on the mass ratio, since both the length and the width of the region are observed to scale with very similar powers of (me/mi). VI. SUMMARY In this paper, we have made scaling arguments based on the balancing of terms in a hyper-resistive Ohm’s law at the boundaries of the electron diffusion region, and have found 10 a scaling for δe equivalent to that found by other recent studies[22, 23, 24] via a somewhat different method. How- ever, we emphasize here that these scaling arguments are ul- timately incomplete unless they make a prediction about the length of the dissipation region. The expression derived for the reconnection rate (Eq. 17) appears to be formally indepen- dent of electron mass, and hyperresistivity. However, formal independenceis no guarantor of fast reconnection, because the upstream field and the length of the electron diffusion region may in fact depend on the dissipation mechanism. If Le is of order di or even smaller then reconnection rate can take on a numerically high value, but numerical results indicate very clearly that Le is a function of ηH for the simple coalescing system under consideration here. The reconnection rate nor- malized to local values of magnetic field and density depends on ηH in this system, which might lead one to conclude that the reconnection is dissipation dependent. However, when the reconnection rate is normalized to typical global values of B and n, it appears that the time scale of the reconnection pro- cess is in fact nearly independent of ηH, and could therefore be termed “fast”. For the purposes of comparison with other systems, these global values in general yield a more mean- ingful value for the normalized reconnection rate, since local values may themselves depend on the dissipation mechanism. Electron inertia does affect the extent of the electron diffu- sion region. However, both the length and width of the elec- tron diffusion region appear to scale in very similar ways, so that the aspect ratio of the region appears to be independent of the mass ratio. The mass ratio dependence of the length of the electron dissipation region found here is in good agree- ment with PIC results. The present study has made no attempt", + "scale in very similar ways, so that the aspect ratio of the region appears to be independent of the mass ratio. The mass ratio dependence of the length of the electron dissipation region found here is in good agree- ment with PIC results. The present study has made no attempt to test the scaling of reconnection with varying guide field, Bz, or boundary conditions, which are left for future work. Additionally, we have considered electron inertia and hyper- resistivity separately in our numerical experiments. However, in the regime where both of these effects are significant, the parametric dependence of the reconnection region on both pa- rameters could be complicated. Recent PIC simulations of reconnection have produced highly elongated current sheets, which break up into many small islands. High Lundquist number simulations in the con- text of resistive MHD have also produced very extended cur- rent sheets, which are subject to a “secondary tearing insta- bility.” This secondary instability also leads to the formation of many small plasmoids, similar to what is seen in the ki- netic systems. The present study was largely motivated by the question of whether these kinds of extended electron cur- rent sheets could be realized within the framework of Hall MHD. Electron viscosity does not on its own appear to be capable of producing highly extended current sheets. In col- lisionless kinetic systems, it has been shown that agyrotropic (off-diagonal) pressure tensor effects, which are not generally included in fluid models, play a dominant role in breaking the frozen-in condition. Hyper-resistivity models some of the physics associated with the electron pressure tensor. It is pos- sible that a more complicated fluid closure parameterizing the physics of the agyrotropic pressure tensor components may be capable of reproducing the current sheet extension and subse- quent plasmoid formation witnessed in kinetic systems. That investigation is a topic of presently ongoing research. 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Wagner1 1Department of Physics, University of Illinois at Urbana-Champaign 2Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853 3Department of Physics, University of Michigan, Ann Arbor, MI 48109 4Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010 5Department of Physics, College of William and Mary, Williamsburg, VA 23185 6IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120, USA 7School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China 8California Institute of Technology, Pasadena, CA 91125 9Department of Physics, King’s College London, Strand, London, WC2R 2LS, U.K 10Department of Chemistry, University of Michigan, Ann Arbor, MI 48104 11Department of Chemistry, University of Colorado, Boulder 12DEN, Service de Recherches de M´etallurgie Physique, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France 13Department of Physics, Columbia University, New York, NY 10027 A large collaboration carefully benchmarks 20 first principles many-body electronic structure methods on a test set of 7 transition metal atoms, and their ions and monoxides. Good agreement is attained between 3 systematically converged methods, resulting in experiment-free reference values. These reference values are used to assess the accuracy of modern emerging and scalable approaches to the many-electron problem. The most accurate methods obtain energies indistinguishable from experimental results, with the agreement mainly limited by the experimental uncertainties. Com- parison between methods enables a unique perspective on calculations of many-body systems of electrons. INTRODUCTION A major challenge in condensed matter physics, ma- terials physics, and chemistry is to compute the proper- ties of electronic systems using realistic Hamiltonians. Efficient and accurate calculations could enable com- putational design of drugs[1] and other materials,[2, 3] and shed light on a number of physical questions, such as the origin of linear-T resistivity,[4] high temperature superconductivity,[5] and many other effects that cur- rently lack satisfying explanation. Many-body quantum calculations on classical comput- ers are challenging because the dimension of the Hilbert space increases dramatically with the number of parti- cles. For example, in the simple case of a CuO molecule with a large (5z) basis, the Hilbert space is of dimension 1044 for the Sz = 1 2 sector. A vector of this size can- not be represented in any computer; at the present time the Oak Ridge machine Summit has approximately 250 petabytes of storage,[6] which is still approximately 17 orders of magnitude too small to store a single vector. Modern techniques therefore use compression and other techniques to approximate the state vectors. There are many, not always mutually exclusive, ap- proaches to dealing with the dimensionality: trunca- tion of the wave function space through wave function ansatzes, one-particle Green function approaches, den- sity", + "small to store a single vector. Modern techniques therefore use compression and other techniques to approximate the state vectors. There are many, not always mutually exclusive, ap- proaches to dealing with the dimensionality: trunca- tion of the wave function space through wave function ansatzes, one-particle Green function approaches, den- sity functional theory, Monte Carlo methods, and em- bedding techniques. The techniques vary dramatically in their computational cost and accuracy. Most studies[7– 15] judge the accuracy of the methods by comparing to experimental energies,[16] which are computed by taking differences of total energies and are therefore subject to fortuitous cancellation of error. Instead, in this study we include 3 systematically improvable methods with suffi- ciently small prefactors that they yield almost exact total energies within the chosen basis set and serve as a bench- mark for testing all other methods. In this manuscript, we apply a diverse array of 20 es- tablished and emerging techniques to a test set of small, realistic transition metal molecules and atoms. Each technique was implemented by an expert, and employed precisely the same Hamiltonian. This approach allows us to directly assess methodological differences without confounders such as different Hamiltonians, and has been important for a previous benchmark study of the hydro- gen chain.[17] For these systems, we achieve convergence of exponentially scaling but systematically convergeable methods at the order of 1 mHartree in the total energy, or about 300 K, establishing a reliable reference on real- istic Hamiltonians with complex atoms. We then assess the accuracy of more approximate approaches for com- puting the total energy of atoms and molecules, which allows some assessment of transferability of performance arXiv:1910.00045v2 [cond-mat.mtrl-sci] 5 Oct 2019 2 with increasing system size. Finally, we study how errors in the total energies translate into errors of physical ob- servables obtained as differences of total energies, and we make comparisons to experiments. These results provide an important reference for the development of techniques that can address the larger goal of computing electronic properties of realistic materials. METHODOLOGY Table I lists the methods tested in this work. It in- cludes most of the common techniques to address the many-electron problem, as well as some emerging meth- ods. It also includes a few methods such as CISD which are no longer commonly used but have historical rele- vance. The methods in this benchmark vary dramati- cally in their computational cost; the density functional theory methods required only a few minutes to complete the test set, while some of the more advanced techniques were not able to treat every basis for every system with the available amount of computer time. The methods also scale very differently, ranging from O(N 3 e ) to expo- nential in the number of electrons Ne. Of the 3 systemat- ically converged methods (iFCIQMC, DMRG and SHCI) only SHCI was performed for all the systems in all the basis sets. Consequently, SHCI energies will be used as the reference. Some of the other techniques are in principle system- atically improvable, such as configuration interaction, coupled cluster, self-energy embedding theory, and the", + "converged methods (iFCIQMC, DMRG and SHCI) only SHCI was performed for all the systems in all the basis sets. Consequently, SHCI energies will be used as the reference. Some of the other techniques are in principle system- atically improvable, such as configuration interaction, coupled cluster, self-energy embedding theory, and the Monte Carlo methods, but convergence to better than 1 mHa was not achieved on these systems for the level of the method employed. Some of the techniques give upper bounds to the exact energy, such as DMC, CISD, DMRG, and HF. Finally, for completeness it should be noted that the methods also require different levels of specification to define the approximations used. For example, some of the methods can be reproduced only by specifying the initial starting determinant; others require defining an initial multideterminantal wavefunction, or the choice of partitioning between high-level and low-level methods. We consider transition metal systems, with the core electrons removed using effective core potentials[51–53]. These potentials accurately represent the core[54] in many-body simulations and allow all the methods con- sidered in this work to use the same Hamiltonian. In addition, they provide an easy way to include scalar rel- ativistic effects, needed for meaningful comparison to ex- periment. These potentials are available for O, Sc, Ti, V, Cr, Mn, Fe, and Cu, which defines our test set. We consider these atoms, their ions, and the corresponding transition metal monoxide molecules. To simplify the comparison, the molecules were computed at their equi- librium geometry. Almost every electronic structure method (all the TABLE I. A list of abbreviations used in this benchmark. De- tails are available in the Supplementary Material. Column A lists the largest basis set employed by that method for at least one of the transition metal atoms, Column B lists the same for the monoxide molecules. The basis sets are abbreviated in order as d, t, q, 5, and c for complete basis set. Abbreviation Method A B AFQMC(MD) Auxiliary field quantum Monte Carlo with a multi-determinant trial function[18, 19] 5 5 B3LYP DFT with the B3LYP functional[20] 5 5 CISD Configuration interaction with singles and doubles 5 5 DMC(SD) Fixed node diffusion Monte Carlo with a single determinant nodal surface[21, 22] c c DMRG Density matrix renormalization group[23, 24] t d GF2 Second order Green function[25, 26] q q HF Hartree-Fock 5 5 HF+RPA Hartree-Fock random phase approximation[27] t t HSE06 DFT with the HSE06 functional[28, 29] 5 5 iFCIQMC Initiator full configuration interaction quantum Monte Carlo[30, 31] q d LDA DFT in the local density approximation[32, 33] 5 5 MRLCC Multireference localized coupled cluster[34–37] 5 5 PBE DFT in the PBE[38] approximation 5 5 QSGW Quasiparticle self-consistent GW approximation[39] t t SCAN DFT with SCAN functional[40] 5 5 SC-GW Self-consistent GW approximation [41, 42] t - SEET(FCI/GF2) Self-energy embedding theory with many-body expansion.[43–47] q q SHCI Semistochastic heatbath configuration interaction[48, 49] 5 5 UCCSD Unrestricted coupled cluster with singles and doubles[50] 5 5 UCCSD(T) Unrestricted coupled cluster with singles, doubles, and perturbative triples[50] 5 5 methods in this study except DMC) works in a", + "t - SEET(FCI/GF2) Self-energy embedding theory with many-body expansion.[43–47] q q SHCI Semistochastic heatbath configuration interaction[48, 49] 5 5 UCCSD Unrestricted coupled cluster with singles and doubles[50] 5 5 UCCSD(T) Unrestricted coupled cluster with singles, doubles, and perturbative triples[50] 5 5 methods in this study except DMC) works in a finite ba- sis. Here, we follow the chemistry convention of defining an ascending basis set denoted by the z(ζ) value, ranging from 2 to 5; i.e., dz, tz, qz, and 5z. For each system, we consider the first principles Hamiltonian projected onto the basis, making for a total of 23 × 4 = 98 calcula- tions for each method. See the Supplementary Material for details on the precise basis sets used in this study. While the results are only comparable to experiment in the complete basis set limit (cbs), for each basis set there corresponds a projected Hamiltonian, which also has an exact solution. We thus can compare methods within a basis since the Hamiltonian is defined precisely. In Table I, we list the methods considered in this work. The deviation in the total energy between two methods m and n is computed as σ(m, n) = sP i∈systems(Ei(n) −Ei(m))2 N , (1) where N is the total number of calculations performed 3 in common between the methods. This is a measure of how well the output total energies between two methods agree. It is possible for two methods with large σ to agree on energy differences if there is significant cancellation of errors. To compare total energies between methods and sys- tems in a consistent way, we use the concept of percent of correlation, commonly used in quantum chemistry: % correlation energy(m) = 100 × EHF −Em EHF −ESHCI , (2) where EHF is the Hartree-Fock energy, m stands for the method under consideration, and ESHCI is the total en- ergy computed in the basis by the SHCI method. At 100% of the correlation energy, the exact result is ob- tained. This quantity is particularly useful since meth- ods tend to obtain similar percentages of the correlation energy across different basis sets and systems. Extrapolation to the basis set limit is done making the usual assumption that the correlation energy (differ- ence between Hartree-Fock and the exact energy) scales as 1/n3, where n is the cardinal number of the basis set, and that the Hartree-Fock energy exponentially con- verges to the complete basis limit. Complete basis set ex- trapolation is necessary for comparison of the finite basis set results to experiment, diffusion Monte Carlo (DMC), and density functional theory results. DMC works di- rectly in the complete basis limit, whereas density func- tional methods are designed to reproduce complete basis set limit energies. The uncertainty in the extrapolation, judged from the variation between different fits to the extrapolation, is approximately 2–4 mHa; for details, see the Supplementary Material. Thus, in this test set, the largest uncertainty in the complete basis set total energy is due to the extrapolation of finite basis set energies to the infinite limit. The energy differences studied", + "variation between different fits to the extrapolation, is approximately 2–4 mHa; for details, see the Supplementary Material. Thus, in this test set, the largest uncertainty in the complete basis set total energy is due to the extrapolation of finite basis set energies to the infinite limit. The energy differences studied are the ionization po- tential of a transition metal atom M: IP = E(M +) − E(M), and the dissociation energy of a metal oxide molecule MO: DE = E(M) + E(O) −E(MO). These quantities have been studied in detail for these systems in the past, for example Refs [7–13, 55–58], among others. However, none of these previous studies have attained reference energies as well-converged as the ones in this paper, and none compare energies from a large number of methods. RESULTS We show several views of the data collected in this study in the figures. The Supplementary Material con- tains various tables and the complete set of data (∼1200 calculations) on which these plots are based. Fig 1 estab- lishes that several high accuracy techniques are in agree- ment and establishes a reference technique SHCI. Fig 2 UCCSD(T) iFCIQMC DMRG SHCI AFQMC(MD) SEET(FCI/GF2) HSE06+RPA QSGW CISD HF+RPA SC-GW GF2 MRLCC UCCSD UCCSD(T) iFCIQMC DMRG SHCI AFQMC(MD) SEET(FCI/GF2) HSE06+RPA QSGW CISD HF+RPA SC-GW GF2 MRLCC UCCSD −3.6 −3.0 −2.4 −1.8 −1.2 Log(RMS) FIG. 1. Cluster analysis of electronic structure methods in this work. The matrix values are the logarithm of the RMS deviation of the total energy in Hartrees (Eq. 1) between the two methods. compares the performance of methods in computing the total energy as compared to the reference. Fig 3 com- pares the performance of methods in computing ioniza- tion potential of the atoms and dissociation energy of the molecules. Fig 4 summarizes the cancellation of error for different techniques in computing the differences in ener- gies. Finally, Fig 5 compares calculations using methods found to be accurate to the experimental dissociation en- ergies. In this section, we examine related methods in the context of these different views. In Fig 1, we show a cluster analysis of the total ener- gies using Eq. 1, evaluated on the intersection of basis sets and systems available for both methods, as the distance metric. iFCIQMC, DMRG and SHCI were converged to very high levels of accuracy. In fact these three methods agree to ∼1 mHa for all systems and basis sets that were computed. Because of this 3-fold agreement, we can take any of these results as the exact ground state energy in a given basis set to within an RMS error of less than 1 mHa, which is approximately what is termed “chemical accuracy” in the context of energy differences. Here we have achieved 1 mHa accuracy in the total energy of the ground state. However, as shown in Table I, iFCIQMC and DMRG calculations were feasible within the avail- able computer time for only the smaller basis sets, so we use SHCI as the reference. For finite basis sets, the es- timated uncertainty is ∼1 mHa, and for the complete basis", + "the ground state. However, as shown in Table I, iFCIQMC and DMRG calculations were feasible within the avail- able computer time for only the smaller basis sets, so we use SHCI as the reference. For finite basis sets, the es- timated uncertainty is ∼1 mHa, and for the complete basis set, the estimated uncertainty is ∼2-4 mHa due to the extrapolation uncertainty. 4 50 60 70 80 90 100 110 120 130 % SHCI correlation energy in basis SEET(FCI/GF2) AFQMC(MD) UCCSD(T) iFCIQMC DMRG SHCI HSE06+RPA QSGW CISD HF+RPA SC-GW GF2 MRLCC UCCSD DMC(SD) LDA PBE HSE06 B3LYP SCAN System type atom molecule 90 95 100 105 110 % SHCI correlation energy in basis FIG. 2. Kernel density estimation[59–61] of the percent of the SHCI-computed correlation energy within each basis obtained by each of the methods in the benchmark set. All basis sets available are plotted; individual data points are indicated by small lines. Density functional methods have a large spread across systems in the percent of correlation energy attained (Fig 2). The gradient corrected and the hybrid func- tionals (B3LYP, HSE06, PBE and SCAN) improve the LDA. The most recently proposed of these, SCAN, is more consistent in the percent of correlation energy ob- tained at around 80–90% of the correlation energy. Fig. 4 shows that it also benefits more than the other function- als from a cancellation of errors between the atom and the molecule to give more accurate dissociation energies, although it has less cancellation of errors for the ioniza- tion potentials. Much of the improvement in accuracy of the hybrid functionals over PBE is in the cancellation of error. The random phase approximation (RPA) and both ver- sions of GW overestimate the correlation energy as shown in Fig 2. While the total energy tends to be too low, those errors tend to cancel for QSGW applied to energy differences, as can be seen in Figs. 3 and 4. As can be seen in Fig 2, configuration interaction with singles and doubles (CISD), a truncated determinant ex- pansion technique well-known to have size consistency defects, performs much better for the atoms than the molecules, which leads to rather poor predictions for the dissociation energy of the molecules (Fig 3). The error is large enough that CISD was not included in Fig 4 to im- prove readability of the more accurate numbers. We note that unrestricted coupled cluster with singles and doubles (UCCSD), which is size consistent, also performs worse on the molecules than the atoms, though to a lesser de- gree than CISD. This results in underestimation of the dissociation energy (Fig 3), and no cancellation of error in the dissociation energy, but significant cancellation in the ionization potential (Fig 4). Fixed node diffusion Monte Carlo with a single deter- minant trial function (DMC(SD)) yields a lower bound to the extrapolated correlation energy, corresponding to an upper bound to the total energy, which is apparent in Fig 2. The remaining energy is the fixed node er- ror, the main approximation in the DMC calculations, which for a single Slater determinant", + "minant trial function (DMC(SD)) yields a lower bound to the extrapolated correlation energy, corresponding to an upper bound to the total energy, which is apparent in Fig 2. The remaining energy is the fixed node er- ror, the main approximation in the DMC calculations, which for a single Slater determinant nodal surface is much larger than the extrapolation uncertainty. With the single Slater determinant, DMC obtains 90–95% of the correlation energy quite consistently, in line with pre- vious benchmarks on smaller systems[62]. This consis- tency results in significant cancellation of error (Fig 4) in the dissociation energy and ionization potential. Self energy embedding theory with a full con- figuration interaction solver and GF2 embedding (SEET(FCI/GF2)) obtains results in good agreement with the reference total energy (Fig 2), resulting in ac- curate energy differences (Fig 3). Consequently, it lies very close to the x = y line in Fig 4 and does not benefit from additional cancellation of error, as the energies are already accurate. The errors in the total energy are not strongly correlated with the atomic species; for example the error in the Ti atom is not statistically similar to the error in the TiO atom, resulting in little cancellation of error. The auxilliary field quantum Monte Carlo with a multi- ple determinant trial function (AFQMC(MD)) gives good agreement with the reference total energy, with an RMS deviations of about 3 to 4 mHa. The dissociation en- ergies have an RMS deviation of ∼2.5mHa, which is 5 −3 −2 −1 0 1 2 Error relative to SHCI in basis (eV) SEET(FCI/GF2) AFQMC(MD) UCCSD(T) iFCIQMC DMRG SHCI HSE06+RPA QSGW CISD HF+RPA SC-GW GF2 MRLCC UCCSD DMC(SD) LDA PBE HSE06 B3LYP SCAN Ionization potential Dissociation energy −0.2 −0.1 0.0 0.1 0.2 Error relative to SHCI in basis (eV) FIG. 3. Kernel density estimation plot of dissociation energy and ionization potential of molecules and atoms to SHCI reference calculations. Methods are ordered according to the clustering in Fig 1. consistent with the conclusion of a recent benchmark on a large set of transition metal diatomics [14]. The use of single determinant UHF trial wave functions would lead to less accurate results, roughly doubling the RMS error in the total energy of the molecules (see Supplementary Material I.A). Coupled cluster with singles, doubles, and perturbative triples (UCCSD(T)) performs very well on these systems, obtaining close to 100% of the correlation energy. For these problems, UCCSD(T) has a notably low cost for high performance. The accuracy of UCCSD(T) is likely due to the fact that these systems are not strongly multi- reference, in that even in the near-exact wave functions, there is a single dominant determinant that makes a large contribution to the wave function. This can be seen by examining the natural orbital occupations; for example in UCCSD, the spin-resolved natural orbitals with large occupations have occupations of 0.96 or greater. The single reference nature also explains the mediocre perfor- mance of the multi-reference methods such as MRLCC, which sacrifice some accuracy in the single reference case to treat multi-reference situations more accurately. In general,", + "example in UCCSD, the spin-resolved natural orbitals with large occupations have occupations of 0.96 or greater. The single reference nature also explains the mediocre perfor- mance of the multi-reference methods such as MRLCC, which sacrifice some accuracy in the single reference case to treat multi-reference situations more accurately. In general, active space techniques, which operate within an explicitly chosen subspace of the larger Hilbert space, are not very effective for these systems. We believe that the reference data produced compu- tationally has lower uncertainties than experiment for the purposes of benchmarking quantum calculation tech- niques. The ionization potential of the large-basis SHCI results is in agreement with experiment with mean abso- lute deviation of 0.2 mHa, or 7 meV, so one could equiva- lently use experiment or the SHCI reference values, as can be verified in Table VI of the Supplementary Material. The experimental dissociation energy estimation is lim- ited by the challenges of the measurements and the exper- imental measures differ from one another by as much as 0.5 eV. In Fig 5, the high accuracy estimates of the disso- ciation energy of the molecules is shown, compared to ex- perimental values with zero point energy removed[63–69]. For these systems, the experimental uncertainty of the dissociation energy is larger than the difference between the most accurate techniques in this benchmark. Re- markably, SHCI, UCCSD(T), and AFQMC(MD) agree to about 0.1 eV for all the molecules. We also should note that since we used effective core potentials to stan- dardize the benchmark, there may be some small errors in comparing directly to experiment. However, we see no evidence that the potentials used are limiting the ac- curacy; the most accurate methods obtain results well within the experimental uncertainty, with the possible exception of VO, for which most of the experimental val- ues are slightly below the theoretical ones. When computing differences of total energies, both methodological errors and errors due to finite basis sets tend to cancel. In Fig 4 we quantify the methodological cancellation of errors in many of the techniques studied in this work. Considering basis set errors, the RMS error in the total energy in the commonly used tz basis com- pared to the complete basis set limit is 75 mHa, while the RMS error in the ionization energy and dissociation energy for the same comparison are 1.6 mHa and 6 mHa respectively, as can be seen in Table VIII in the Supple- mentary information. 6 0 50 100 150 200 250 Total energy RMS errors (mHa) 0 5 10 15 20 Ionization potential RMS errors (mHa) AFQMC(MD) B3LYP DMC(SD) HF+RPA HSE06 HSE06+RPA LDA MRLCC PBE QSGW SC-GW SCAN SEET(FCI/GF2) UCCSD UCCSD(T) a 0 50 100 150 200 250 Total energy RMS errors (mHa) 0 10 20 30 40 50 60 Dissociation energy RMS errors (mHa) AFQMC(MD) B3LYP DMC(SD) HF+RPA HSE06 HSE06+RPA LDA MRLCC PBE QSGW SCAN SEET(FCI/GF2) UCCSD UCCSD(T) b FIG. 4. Cancellation of error for many methods in this study, computed by comparing the RMS error in the total energy to the RMS error in the (a)", + "50 60 Dissociation energy RMS errors (mHa) AFQMC(MD) B3LYP DMC(SD) HF+RPA HSE06 HSE06+RPA LDA MRLCC PBE QSGW SCAN SEET(FCI/GF2) UCCSD UCCSD(T) b FIG. 4. Cancellation of error for many methods in this study, computed by comparing the RMS error in the total energy to the RMS error in the (a) ionization energy of the atoms and (b) dissociation energy for molecules. HF and CISD were excluded from the comparison for more detail in the more accurate methods; they are offthe scale here. The red dashed line corresponds to no cancellation of error. CONCLUSION We surveyed 20 advanced many-electron techniques on precisely defined realistic Hamiltonians for transition metal systems. For a given basis set, we achieved ∼1 mHa agreement on the total energy between high ac- curacy methods, which provides a total energy bench- mark for many-body methods. To our knowledge, such an agreement is unprecedented for first principles calcu- lations of transition metal systems. Our accurate refer- ence energies should enable the development of approx- imate, but more computationally efficient, many-body techniques as well as better density functionals, with- out the necessity of experimental reference values. These systems are also a useful test for future quantum comput- 4.00 4.25 4.50 4.75 5.00 CrO AFQMC(MD) SHCI UCCSD(T) 2.6 2.8 3.0 CuO 4.0 4.2 FeO 3.50 3.75 4.00 4.25MnO 6.8 6.9 7.0 Dissociation energy (eV) ScO 6.6 6.8 7.0 TiO 1960 1970 1980 1990 2000 2010 2020 Year 6.2 6.4 6.6 VO FIG. 5. Comparison of 5z dissociation energies of the transi- tion metal molecules obtained from the more accurate meth- ods used in this work to experiment. The x-axis is the year the experimental result was published and the width of the bars indicate statistical/systematic uncertainties. ing algorithms. To enable such comparisons, we include pyscf scripts that can execute the benchmark for any density functional available in libxc[70], and can export the one- and two-body integrals needed for testing many- body methods. We have assessed the state of the art in achieving high accuracy in realistic systems. The benchmark set in- cludes systems with large Hilbert spaces of around 1044 determinants. While these spaces are so large that a 7 single vector cannot fit in any computer memory, the computations are feasible due to powerful compression of that space. The systematically converged techniques used in this work (DMRG, FCIQMC, and SHCI), were able to achieve excellent agreement, but can be applied only to relatively small systems due to their computa- tional cost. It is thus important to understand the errors in lower-scaling techniques that can be applied to larger systems, and whether performance on small systems is transferable to larger systems. Our study takes a step in that direction, since we were able to achieve converged results for both correlated atoms and molecules, and in- deed we observed that the accuracy of some techniques degrades with system size. To avoid misinterpretation of the results, we make a comment here. In order to ensure high quality results, it was necessary to limit the number of systems on which this benchmark was performed. While treating", + "and in- deed we observed that the accuracy of some techniques degrades with system size. To avoid misinterpretation of the results, we make a comment here. In order to ensure high quality results, it was necessary to limit the number of systems on which this benchmark was performed. While treating electron correlation accurately is important to obtain accurate re- sults, these systems have a particular character of corre- lation. 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F. Dalleska, and P. B. Armentrout, The Journal of Chemical Physics 95, 7263 (1991). [70] M. A. L. Marques, M. J. T. Oliveira, and T. Burnus, Computer Physics Communications 183, 2272 (2012). 1 I. DETAILED METHODOLOGY A. AFQMC The AFQMC method [1–3] estimates the ground-state properties of a many-fermion system by statistically sampling the wave function |ψg⟩∝e−β ˆ H|ψ0⟩, where |ψ0⟩is an initial wave function which is nonorthogonal to the ground state. The projection is carried out iteratively by small time step e−∆τ ˆ H, with β = n∆τ sufficiently large to project out all excited states. The propagator is represented as e−∆τ ˆ H = R dxp(x) ˆB(x) where ˆB(x) is a one-body operator which depends on the vector x, and p(x) is a probability distribution. This representation maps a many-body system into an ensemble of one-body systems, with the ensemble then sampled by Monte Carlo (MC) techniques. We use open- ended random walks in Slater determinant space to sample the imaginary time projection and represent the ground state wave function: |ψg⟩= R dφ cφ|φ⟩, where the Slater determinants in the integral are non-orthogonal ⟨φ′|φ⟩̸= 0. A gauge constraint, implemented approximately with a trial wave function |ψT ⟩, is applied on the sampled Slater determinants [1, 2] to control the sign or phase problem. In this work, we present results obtained from the AFQMC method implemented for Gaussian basis sets [4, 5]. We set the linear dependence threshold to be 10−8 for the one-electron basis [6] and use the modified Cholesky decomposition [7] with a threshold 10−6 for the Coulomb interaction. Most of the calculations use projection time β = 35 E−1 Ha and time step ∆τ = 0.005 E−1 Ha. The convergence error from finite β is negligible and extrapolations are performed when the Trotter error is larger than Monte Carlo uncertainty. The reported error bars are estimated by one standard deviation statistical errors. Truncated CASSCF wave functions were used as |ψT ⟩here. Fast update procedure [8, 9] allows the use of multi- determinant CAS trial wave functions with sublinear cost. (For example, in ScO TZ basis, the cost of the AFQMC calculation with a |ψT ⟩of 163 determinants is 2.1× that of a single determinant calculation.) Typically around 10 CAS orbitals are used to generate the |ψT ⟩in both atoms and molecules. Following procedures in past AFQMC calculations using CASSCF [10, 11], we truncate the wave function by discarding determinants with the smallest weights, up to an integrated weight of δ = 10−3, which results in ∼100 determinants in most cases. Because of the fast update algorithm, we could check the effect on the AFQMC results of increasing the CAS space to the next level with little cost. In VO and FeO, a noticeable difference was seen outside the", + "of δ = 10−3, which results in ∼100 determinants in most cases. Because of the fast update algorithm, we could check the effect on the AFQMC results of increasing the CAS space to the next level with little cost. In VO and FeO, a noticeable difference was seen outside the statistical error, and we increased the CAS space to 12, resulting in ∼1700 and ∼1600 determinants, respectively, in their |ψT ⟩. In solids, CASSCF trial wave functions would not be applicable straightforwardly in a size-consistent manner. There have been many benchmark studies of AFQMC using single determinant trial wave functions (e.g. Refs. [10–12]). To give an idea of the dependence of the constraint error in AFQMC for the specific systems here, the computed total energies in TiO, VO, CrO, MnO change by about −4.9, −1.5, +6.8, and −9.7 milli-Hartrees from the reported multi-det |ψT ⟩results when a single determinant UHF trial wave function is used (TZ basis, with MC error bars 1-2 milli-Hartrees). Besides using a single Slater determinant, there are a number of possibilities to systematically improve the trial wave function, including generalized Hartree-Fock [13], symmetry projection and symmetry-adapted multi-determinants [6, 14, 15], Hartree-Fock-Bogoliubov form [16, 17], self-consistent trial wave functions [18, 19]. B. Configuration interaction Configuration interaction with singles and doubles excitations (CISD) was used as implemented in the PySCF package. CISD approximates the many-body wave function as a sum of Slater determinants, constructed from a reference Slater determinant, which was taken from restricted open shell Hartree-Fock. In CISD, the wave function is given as |ΨCISD⟩= (c0 + X ij,σ C(s) ai c† a,σci,σ + X ijkl,σ,σ′ C(d) abij,σ,σ′c† a,σc† b,σ′ci,σcjσ′ )|ΨHF ⟩, where c† and c are creation/destruction operators, respectively, a, b refer to virtual orbitals, i, j refer to occupied orbitals, and all C parameters are variationally optimized. CISD scales approximately as O(N 6 e ) and is known to not arXiv:1910.00045v2 [cond-mat.mtrl-sci] 5 Oct 2019 2 TABLE I. Calculations in the database. In each cell, the symbols correspond to the basis performed as follows: d: vdz, t: vtz, q: vqz, 5: v5z, c: cbs. A dash means that there were no calculations for that system using that technique. System Method Cr Cr+ CrO Cu Cu+ CuO Fe Fe+ FeO Mn Mn+ MnO O O+ Sc Sc+ ScO Ti Ti+ TiO V V+ VO AFQMC(MD) dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 B3LYP dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 CISD dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 DMC(SD) c c c c c c c c c c c c c c c c c c c c c c c DMRG dt dt d dt dt d dt dt d dt dt d dt dt dt dt d dt dt d dt dt d GF2 dtq dtq dtq", + "c c c c c c c c c c c c c c c c c c c c c c DMRG dt dt d dt dt d dt dt d dt dt d dt dt dt dt d dt dt d dt dt d GF2 dtq dtq dtq — — dtq dtq dtq dtq dtq dtq dt dtq dtq dtq dtq dtq dtq dtq dtq dtq dtq dtq HF+RPA t t t t t — dt t t t t t dtq dtq t t t t t t t t t HSE06 dtq5 dtq5 dtq5 dtq5 dtq5 — ddtq5 dtq5 dtq5 dtq5 dtq5 dtq5 ddtq5q ddtq5q dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 LDA dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 MRLCC dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 PBE dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 PBE+RPA t t t t t — t t t t t t dtq dtq t t t t t t t t t QSGW t t t — — — dt t t t t t dtq dtq t t t t t t t t t SC-GW d d — — — — d d — d d — d d d d — d d — d d — SCAN dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 SEET(FCI/GF2) dtq dtq dq — — dtq dtq dtq dtq dtq dtq — dtq dtq dtq dtq dtq dtq dtq dtq dtq dtq dtq SHCI dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 UCCSD dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 UCCSD(T) dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 dtq5 iFCIQMC dtq dtq — dtq dtq — dtq dtq — dtq dtq — dtq dtq dtq dtq d dtq dtq — dtq dtq — 3 be size extensive. C. Coupled Cluster Unrestricted coupled cluster was used as implemented in the PySCF package.[20] The reference state was restricted open shell Hartree-Fock. We found that an unrestricted reference state led to worse results by values up to 10 mHa for the total energy of the molecules. In UCCSD, the wave function is approximated as |ΨCCSD⟩= e ˆT |ΨHF ⟩, (1) where the ˆT operator contains one and two body operators. The exponential ansatz ensures that the technique is size extensive in contrast to CISD. UCCSD scales approximately as O(N 6 e ) [21]. UCCSD(T) evaluates the perturbative effect of including three-body operators", + "approximated as |ΨCCSD⟩= e ˆT |ΨHF ⟩, (1) where the ˆT operator contains one and two body operators. The exponential ansatz ensures that the technique is size extensive in contrast to CISD. UCCSD scales approximately as O(N 6 e ) [21]. UCCSD(T) evaluates the perturbative effect of including three-body operators from a UCCSD reference, and is often called the gold standard of quantum chemistry when used for equilibrium properties. It scales approximately as O(N 7 e ). Despite the steep formal scaling, the prefactor is quite small. Thus compared to the other accurate methods in this paper, namely, SHCI, DMRG, FCIQMC, AFQMC, and SEET, the UCCSD(T) calculations were the least expensive by a significant amount. D. Density Functional Theory Density functional theory (DFT) in the restricted open shell Kohn-Sham approach was used as implemented in the PySCF package.[20] Level 6 grids were used to improve the accuracy, and the resultant state was carefully checked to ensure that it was the DFT ground state, since often the self consistent field process converged to the incorrect state. The basis set error in DFT is very small, typically with less than 1 mHartree difference between the vtz and v5z basis sets. This is because the basis in DFT only has to express the occupied Kohn-Sham orbitals accurately, and is not used to describe electron correlation. Strictly speaking, the DFT energy is only comparable to the many-body solution in the complete basis set, because the functionals are designed to approximate the correlation energy in the basis set limit. E. DMRG The density matrix renormalization group (DMRG) [22] provides a variational ansatz for the wavefunction of the matrix product state form, |Ψ⟩= X n1n2...nk [An1An2 . . . Ank]11|n1n2 . . . nk⟩ (2) where An is a matrix of variational parameters for each orbital and |n1n2 . . . nk⟩is an occupancy vector. The above ansatz expresses the coefficient of any occupancy vector as a product of matrices, where the “bond” dimension of the matrix An is M × M. M may be increased until the ansatz is exact, which happens for M approximately the square root of the full Hilbert space size. The cost of the calculation using the quantum chemistry Hamiltonian with quartic interactions is proportional to M 3k3 + M 2k4 where k is the number of orbitals [23–26]. In a localized basis, the M required for a given accuracy scales like eV D/D+1 where V is the volume of the system and D is the dimension [27]. Thus, when extending a system along one dimension, M is independent of system size, while when extending a system along all three dimensions, the computational scaling is eV 2/3. In any dimension, this is therefore a savings over full configuration interaction, which scales like eV . As can be seen, the ansatz requires an ordering of the orbitals and also treats all orbitals on the same footing. The latter means that in practice the DMRG is often a good ansatz relative to many methods when there are active orbitals which needed to be treated in", + ". As can be seen, the ansatz requires an ordering of the orbitals and also treats all orbitals on the same footing. The latter means that in practice the DMRG is often a good ansatz relative to many methods when there are active orbitals which needed to be treated in a balanced way. However, it is inefficient when there are many doubly occupied or empty orbitals. The atoms and molecules in this system fall into this latter single-reference category. Thus we do not expect the DMRG to be especially efficient, but it serves as a near-numerically exact method to benchmark other techniques more suitable for these systems. 4 The DMRG calculations in this work were carried out using a spin-adapted code (a slight modification of the above ansatz) which allows us to obtain pure spin states [28]. The orbital ordering was generated by the default genetic algorithm [29]. We used the two-site variant of the DMRG and carried out calculations systematically increasing M. The largest M we used ranged from 4000 - 10000. To verify the accuracy of the energy we carried out an extrapolation in the total energy. We did this either by the standard linear extrapolation in the energy against the discarded weight in the two-site algorithm, where the DMRG energies at different M were computed by sweeping backwards from the largest M down to smaller M’s (backwards schedule) [24, 29, 30], or by extrapolating the energies of the largest M values against 1/M. In the first case, the extrapolation error is usually reported as a fraction of the extrapolation distance between the lowest variational energy and the extrapolated energy. This extrapolated energy was consistent with the energy obtained by extrapolating against 1/M, but in some cases the 1/M extrapolation was more linear, and we report that as the extrapolated energy. The DMRG energies at the largest M are variational. Where included, they provide the lowest variational energies for this benchmark. F. FCIQMC The FCIQMC method [31–33] directly samples the many body wavefunction by stochastic propagation of a popula- tion of discrete walkers in Slater determinant space, defined by a given single particle basis. The annihilation of walkers with anti-walkers is crucial to ensure that the average wavefunction has the correct configurational sign-structure, and can lead to a circumvention of the Fermion sign problem without uncontrolled approximations. When a determinant is occupied by a small number of walkers, it is not clear whether the determinant should be ultimately dominated by walkers or anti-walkers, and so spawning new walkers from such a determinant can cause incorrect sign information to propagate throughout the network. This problem is minimized by the systematically improvable approach of Initiator FCIQMC [34, 35], which only allows the creation of new walkers on currently unoccupied determinants, by parent walkers residing on a determinant with a population above some threshold (in this work taken to be 3 walkers). This increases the incidence of annihilation events and encourages the sign-coherent propagation of walkers, and results in a convergence to the exact wavefunction energy and properties as the number of walkers", + "walkers residing on a determinant with a population above some threshold (in this work taken to be 3 walkers). This increases the incidence of annihilation events and encourages the sign-coherent propagation of walkers, and results in a convergence to the exact wavefunction energy and properties as the number of walkers increases. Furthermore, small subspaces are identified to define a ‘trial wavefunction’ onto which the sampled wavefunction is projected to calculate the energy, as well as another small subspace in which exact propagation can occur (the semi-stochastic adaptation [36, 37]). These subspaces serve to minimize the stochastic errorbars of the estimators. All FCIQMC calculations in this work were undertaken via the following process: • A maximum walker population Nwalker is chosen • The walker population is initialized on a single determinant and then allowed to reach Nwalker • A short interval in imaginary time after the maximum population is reached, the trial wavefunction space is initialized by performing an exact deterministic diagonalization of the Hamiltonian in the subspace spanned by the NTWF most populated determinants (NTWF ∼200 determinants in this work) • At the same iteration, a subspace of the NSS most populated determinants is identified and designated as the semi-stochastic space. Thereafter, the walkers residing in this subspace are exactly propagated. (NSS ∼10,000 determinants in this work) • The walker population is left to evolve under initiator FCIQMC dynamics until the numerator and denominator of the trial wavefunction projected energy stabilize around a mean value • The energy is taken to be the ratio of means of the numerator and denominator of the energy • The stochastic error is estimated using the Flyvberg – Peterson analysis[38] for serially correlated data. The stochastically sampled wavefunction is affected by the systematic error introduced by the initiator criterion for spawning. In order to reduce this error to within acceptable bounds, the above procedure is repeated for increasingly large values of Nwalker until convergence of the energy estimate with respect to this parameter is achieved. This was approximately 15 million, 50 million, 100 million and 200 million walkers for the Tm/Tm+ vdz, vtz, vqz and TmO vdz calculations respectively[39]. This computational effort very roughly corresponds to 100 core hours per million walkers, with a maximum of ∼15,000 CPU hours used in order to converge to ScO (vdz) system to small random error bars (200 million walkers). 5 G. Fixed node diffusion Monte Carlo Fixed node diffusion Monte Carlo was used as implemented in the QWalk package.[40] A single determinant was generated using PySCF density functional theory in the B3LYP approximation. This determinant gave the lowest upper bound energy to the ground state. We multiplied the determinant by a 3-body Jastrow factor, which then was optimized to minimize the total energy using the linear method.[41, 42] The resultant single determinant Slater- Jastrow wave function was used as a guiding function for diffusion Monte Carlo.[43] In this method, the diffusion Monte Carlo wave function is given by |ΨDMC⟩= e−τ ˆ H|ΨT ⟩, (3) where |ΨT ⟩is the trial wave function, in this case the Slater-Jastrow wave function. The", + "resultant single determinant Slater- Jastrow wave function was used as a guiding function for diffusion Monte Carlo.[43] In this method, the diffusion Monte Carlo wave function is given by |ΨDMC⟩= e−τ ˆ H|ΨT ⟩, (3) where |ΨT ⟩is the trial wave function, in this case the Slater-Jastrow wave function. The T-moves scheme[44] was used to ensure an upper bound to the ground state energy. Timestep errors were extrapolated out using a linear fitting process and timesteps as low as 0.0025 Hartrees−1. Scaling in stochastic methods is complex. To obtain the total energy with a given stochastic uncertainty, DMC(SD) scales as O(N 3 e + ϵN 4 e ). The N 4 e cost typically does not appear until the number of electrons is more than 400-500. This method has been applied on systems with more than 1000 electrons. We also considered more accurate trial wave functions, which can improve the fixed node error. We constructed them from SHCI wave functions by running small selected CI, and choosing the determinants with the largest weights. These weights were then reoptimized in the presence of the Jastrow factor using the linear method.[41, 42] H. GF2 The fully self-consistent second order Green’s function theory (GF2) [45–49] includes all second-order skeleton diagrams dressed with the renormalized second-order propagators and bare interactions. GF2 is formulated as a low-order approximation to the exact Luttinger-Ward (LW) functional [50] and therefore is Φ-derivable, thermody- namically consistent, and conserving [51, 52]. For transition atoms, we solve all the non-linear equations self-consistently at non-zero temperature. At each iteration, the self-energy, Green’s function, and Fock matrix are updated until convergence is reached, so that the converged solution is reference-independent. Because of stability problems in the self-consistency, the calculations for transition monoxides are only done with one-shot GF2 on top of unrestricted Hartree-Fock. The self-energy, Green’s function, and Fock matrix are not iterated until self-consistency. All the calculations are done on a mesh that combines a sparse power-law grid with an explicit transform based on a Legendre expansion of the self-energy. [53] I. MRLCC The multi-reference linearized coupled-cluster (MRLCC) is a flavor of multi-reference perturbation theory. We consider a reference wavefunction |Ψ0⟩obtained for example from a CAS-like calculation. The out-of-active-space dynamical correlation may be added by multi-reference perturbation theory, where the expressions of the four first contributions to the energy are E0 = ⟨Ψ0| ˆH0|Ψ0⟩ (4) E1 = ⟨Ψ0| ˆV |Ψ0⟩ (5) E2 = ⟨Ψ0| ˆV |Ψ1⟩ (6) E3 = ⟨Ψ1| ˆV −E1|Ψ1⟩ (7) and where the first order correction to the wavefunction |Ψ1⟩obeys (E0 −ˆH0)|Ψ1⟩= ˆV |Ψ0⟩. (8) In Rayleigh-Schr¨odinger perturbation theory, the partitioning of the Hamiltonian ˆH = ˆH0 + ˆV is so that |Ψ0⟩ and E0 are the eigenvector and eigenvalue of the zeroth order Hamiltonian ˆH0. As is well known, this leaves some flexibility for the choice of ˆH0, leading to different perturbation theories having different properties: the use of the 6 Fock operator yields CASPT[54], and the use of the Dyall Hamiltonian[55] yields NEVPT[56, 57]. In this work, we show derivations and results for the MRLCC perturbation theory[58–61], which uses", + "leaves some flexibility for the choice of ˆH0, leading to different perturbation theories having different properties: the use of the 6 Fock operator yields CASPT[54], and the use of the Dyall Hamiltonian[55] yields NEVPT[56, 57]. In this work, we show derivations and results for the MRLCC perturbation theory[58–61], which uses the Fink Hamiltonian[62, 63]: ˆH0 = \u0010 X mn tn m ˆEn m + X mnop vop mn ˆEop mn \u0011 ∆=0, (9) where the spin-free excitation operators are written with a hat, t and v are tensors, and the m, n, o, p indices refer to any molecular orbital. The notation ∆= 0 indicates that only terms that do not change the number of electrons in the core, active and virtual spaces are taken into ˆH0. It follows that: ˆV = \u0010 X mn tn m ˆEn m + X mnop vop mn ˆEop mn \u0011 ∆̸=0, (10) and one can readily see from Eqs. (4) and (5) that the zeroth order energy is the energy of the reference wavefunction and that E1 = 0, which goes to say that this zeroth order Hamiltonian is somewhat close to the exact ˆH and is good in the context of perturbation theory (this is also the case for NEVPT). In the internally contracted scheme, the first order correction to the wavefunction is expressed as a sum over eight class contributions |Ψc 1⟩expanded on perturber wavefunctions that are connected to the reference wavefunction: |Ψc 1⟩= X I dc I ˆEc I|Ψ0⟩, (11) with coefficients dc. In this representation the Fink Hamiltonian is block diagonal (and so is the Dyall Hamiltonian), and the coefficients dc in Eq. (11) are found by solving Eq. (8) subsequently for each class. Projection of Eq. (8) onto the basis of a class yields: Acdc = Scwc, (12) with Ac IJ = ⟨Ψ0| ˆEc I † h (E0 −ˆH0), ˆEc J i |Ψ0⟩ (13) Sc IJ = ⟨Ψ0| ˆEc I † ˆEc J|Ψ0⟩ (14) where realizing that (E0 −ˆH0)|Ψ0⟩= 0 allows the introduction of the commutator (see for example Ref. 61). The terms to manipulate to compute A, S and E2, E3 involve long strings of creation/annihilation operators, and one can use the Wick’s theorem to simplify the expressions. This will result in series of tensor contractions involving one and two electron integrals (stemming from ˆH0 and ˆV ), and RDMs up to fourth order. The application of the Wick’s theorem to the strings of operators is done with the “Second Quantization Algebra” symbolic algebra Python library [? ], which we modified to fit our needs. The scripts also automate the generation of the C code used to solve the resulting equations and to calculate the energies E2 and E3. J. QSGW Typically RPA total energies are done from a simple one-body reference Hamiltonian H0 such as PBE, HSE06, or Hartree-Fock. Since different choices yield different results, there are significant and unavoidable ambiguities. First of all, there is a fundamental issue: When employing a non-self-consistent Green’s function, the different formulas for the total energy do differ in practice: the Galitskii-Migal", + "simple one-body reference Hamiltonian H0 such as PBE, HSE06, or Hartree-Fock. Since different choices yield different results, there are significant and unavoidable ambiguities. First of all, there is a fundamental issue: When employing a non-self-consistent Green’s function, the different formulas for the total energy do differ in practice: the Galitskii-Migal and the RPA total energies are not equal any more [64]. The starting-point dependence can be surmounted by iterating G to self-consistency, that is, by finding a G generated by GW that is the same as the G that generates it (Gout=Gin). But it has long been known that full self-consistency can be quite poor in solids [65, 66]. A recent re-examination of some semiconductors [67] confirms that the dielectric function (and concomitant QP levels) indeed worsen when G is self-consistent, for reasons explained in Appendix A in Ref. [68]. Fully scGW becomes more problematic in transition metals [69]. Finally, scGW is a conserving approximation in the Green’s function G, but W loses its usual physical meaning as a response function. An alternative is the Quasiparticle Self-Consistent GW approximation[70] (QSGW ). It is similar to scGW, but at each cycle the dynamical self-energy is rendered static and hermitian, forming a new noninteracting G0 by making the substitution V xc = 1 2 X ij |ψi⟩{Re[Σ(εi)]ij + Re[Σ(εj)]ij} ⟨ψj|, (15) 7 where Re{} stands for the Hermitian part of the operator. This process is carried through to self-consistency. It was formally justified [68, 70] as a construction to optimize the noninteracting Green’s function G0 by minimizing some measure of the difference |G −G0|. More recently it has been justified as minimizing the the gradient of the Baym-Kadanofffunctional in the space of all G0 [71]. QSGW is nevertheless an approximate self-consistent procedure, which relies on effective one-electron wavefunctions instead of the full Green’s function. As a consequence, the difference between the expressions for the total energy (Galitskii-Migdal or RPA) still persists once the self-consistency has been reached. In this work, we defined the QSGW total energy as the one obtained from the RPA expression based on QSGW eigenvalues and wavefunctions, consistently with Ref. 72. Indeed, the RPA total energy, through the adiabatic connection, is capable of incorporating the correlated part of the kinetic energy [73], whereas the Galitskii-Migdal is not. The correlated part of the kinetic energy is sizeable and it is important to include it properly. It is well known that GW overestimates the correlation energy. A prior study of weakly correlated molecular dimers[74] showed that (1) the RPA tends to systematically overestimate the correlation energy, and (2) the error is connected with short-ranged correlations. This tendency is also found here, as noted in the main text. The ionization energy and the dimer formation energy, both of which benefit from partial cancellation of errors in short-range correlation, are much better described. We also find that the RPA total energy based on QSGW, with its optimal choice for G0, perform significantly better than RPA based on other G0, e.g. PBE or Hartree Fock, as will be shown elsewhere. Finally, it is has been established,", + "errors in short-range correlation, are much better described. We also find that the RPA total energy based on QSGW, with its optimal choice for G0, perform significantly better than RPA based on other G0, e.g. PBE or Hartree Fock, as will be shown elsewhere. Finally, it is has been established, using less optimal forms for G0, that low-order diagrammatic corrections (es- pecially second order screened exchange) significantly reduce errors in the correlation energy [75, 76]. It is shown elsewhere [77] how ladders dramatically improve the dielectric function in TM oxide crystals, so it is reasonable to expect that correlation energies computed from it will see a similar improvement. Density-functional approximations for the exchange-correlation kernel significantly improve on heats of formation of dimers from sp elements [78]. K. RPA We calculated the RPA total energy using the Tamm-Dancoffapproximation (specifically Eq. (9) from Ref. [79]), using several flavors of G0: PBE, Hartree-Fock, HSE06, and QSGW, for the TM atom and M+O dimer. All the RPA calculations in this paper use the MOLGW code [80]. L. sc-GW The GW method [81] evaluates a subset of terms of a diagrammatic weak coupling series in the interaction V deterministically. The GW approximation can be understood as a first-order approximation to Hedin’s series of renormalized propagators and interactions. It is expressed in terms of self-energies Σ, Green’s functions G, screened interactions W and polarizations P by the self-consistent solution of the equations G = G0 +G0ΣG, W = V +V PW, where G0 denotes the Hartree-Fock Green’s function, and Σ and P are computed as Σ = −GW and P = GG. The main difference to GF2 is that, in GW, both propagators and interactions are renormalized, whereas GF2 only renormalizes the propagators. However, GF2 obtains all second order contributions, whereas the second order exchange is missing from GW. Our results are converged to self-consistency, using a finite-temperature imaginary time formulation evaluated at temperatures low enough that the system is in its ground state. Sparse imaginary time and Matsubara frequency grids based on Chebyshev polynomials and the intermediate representation (IR) [82–84] is used, which significantly reduced the computational cost. Our code is based on the ALPS libraries [85, 86]. Self-consistent GW is a Φ−[51, 52] and Ψ-derivable [87] weak coupling method, in the sense that it neglects some diagrams of order V 2. Achieving full self-consistency requires the storage and manipulation of W, which is a frequency-dependent four-index tensor. The necessity of handling this object numerically restricts the method in our implementation to relatively small system sizes. M. Self-energy embedding theory (SEET) SEET [88–95] is a finite temperature Green’s function embedding method. The embedding construction allows us to describe the weakly and strongly correlated orbitals at different levels of theory. The weakly correlated orbitals are 8 treated by a low level, most often a perturbative method (here Green’s function second order (GF2) [46–48, 96–98] or a single iteration of GF2). The strongly correlated orbitals are treated with a high level, usually non-perturbative method. When multiple strongly correlated orbitals are present, they are separated into several intersecting or non", + "a low level, most often a perturbative method (here Green’s function second order (GF2) [46–48, 96–98] or a single iteration of GF2). The strongly correlated orbitals are treated with a high level, usually non-perturbative method. When multiple strongly correlated orbitals are present, they are separated into several intersecting or non intersecting subsets Ai. Each of these subsets contains M A i orbitals and M = P i M A i + M R, where M is the total number of orbitals in the problem and M R are all the orbitals that are not contained in the groups of the strongly correlated orbitals. The orbitals from each of the subsets Ai are used to construct Anderson Impurity Models (AIM) that are then solved by a non-perturbative method, here full configuration interaction (FCI) [99–101]. The intersubset interactions are treated most commonly at a perturbative level. Orbitals chosen to each of the Ai subsets can be chosen based on several criteria such as occupancies of natural orbitals (NOs) or energies of molecular orbitals (MOs), for details see Refs. [48,50]. A general SEET functional can be written as ΦSEET MIX = Φtot weak + P( n k) i (ΦAk i strong −ΦAk i weak) (16) ± Pk=1 k=K−1 P( n k) i (ΦBk i strong −ΦBk i weak), where Φtot weak, in this work is a GF2 solution for the whole orbital space, ΦAk i strong is obtained from the solution of AIM for the strongly correlated subset of orbitals Ai, ΦAk i weak is the solution of the subset Ai with a weakly correlated method used to remove the double counting. The terms (ΦBk i strong −ΦBk i weak) are present in case of intersecting subsets Ai and are necessary to remove the double counting, for details see Ref. 93. We denote a particular SEET calculation as SEET(method strong/method weak)-m([MAo]/basis) since self-energies from intersecting orbital subspaces containing M A orbitals are treated with “method strong”. The whole system is treated with “method weak” and the orbitals from subsets Ai are transformed to a certain orbital basis denoted here as “basis”. In this paper, we most commonly use the basis of molecular orbitals. The details of the finite temperature imaginary time GF2 grid as well as the frequency grid can be found in Ref. 53, 102, and 103. N. SHCI The semistochastic heat-bath configuration iteration (SHCI) method [104–107], is an efficient instance of the general class of methods wherein a selected configuration interaction is performed followed by a perturbative correction (SCI+PT). SCI+PT methods have two stages. In the first stage a set of “important” determinants are selected the Hamiltonian is diagonalized in the subspace of these determinants, V, to obtain the the lowest few eigenstates (or the lowest state if one is interested in the ground state only). In the second stage, a second-order perturbation theory is used to calculate the energy contributions of all determinants that do not belong to the space V but have a non-zero Hamiltonian matrix element with at least one of the determinants in V. Such methods have been used", + "ground state only). In the second stage, a second-order perturbation theory is used to calculate the energy contributions of all determinants that do not belong to the space V but have a non-zero Hamiltonian matrix element with at least one of the determinants in V. Such methods have been used for about 50 years [108–110] and continue to be a subject of interest [111–119] to the present day. We briefly describe the two innovations that account for the time- and memory-efficiency of SHCI. A more detailed description can be found in 107. 1. During the variational and the perturbative steps, straightforward implementations of SCI+PT scan all deter- minants connected to at least one of the determinants in V and select those determinants (|Da⟩) for which the absolute value of the 2nd-order perturbative contribution to the energy \u0000P Di∈V Haici \u00012 EV −Ea > ϵ, (17) where the subscript a denotes a determinant not currently present in V, EV is the energy of the current variational wavefunction, Ea is the energy of determinant Da, and ϵ is a parameter that controls the number of determinants selected. During the variational stage, this is done iteratively to build up the variational wavefunction starting from a single determinant. During the perturbative stage, ϵ = 0. Instead, SHCI modifies the selection criterion to [104], max Di∈V |Haici| > ϵ, (18) which greatly reduces the cost by taking advantage of the fact that most of the matrix elements, Hai, are 2-body excitations, which depend only on the indices of the 4 orbitals whose occupations change and not on the other 9 occupied orbitals of a determinant. Thus by presorting the absolute values of all possible matrix elements of the 2-body excitations in descending order, the scan over determinants Da can be terminated when |Hai| drops below ϵ/ci. A similar idea is used to speed up the selection of 1-body excitations as well. This enables a procedure in which only the important determinants which will be included in the variational wavefunction, or make significant contributions to the perturbative correction, are ever looked at, resulting in orders of magnitude saving over a naive implementation of the SCI+PT algorithm! Different values of ϵ are used during the variational and the perturbative stages of the calculation, which we denote by ϵ1 and ϵ2. 2. The first innovation greatly speeds up both the variational and the perturbative steps of the algorithm. However, the perturbative step has a very large memory requirement when V has a large number of determinants (say 109) because all determinants that are connected to those in V must be stored. [120] We have developed a 2-step [105] and later a 3-step [107] semistochastic perturbative approach that both completely overcomes this memory bottleneck and is faster than the deterministic approach. In addition to these two major methodological improvements, the SHCI method uses auxiliary arrays to speed up the computation of the Hamiltonian matrix [107]. Further, it makes extensive use of hashing and techniques such as variable-byte encoding, hardware atomic operations, dynamic load-balancing and thread pooling to achieve a high efficiency", + "In addition to these two major methodological improvements, the SHCI method uses auxiliary arrays to speed up the computation of the Hamiltonian matrix [107]. Further, it makes extensive use of hashing and techniques such as variable-byte encoding, hardware atomic operations, dynamic load-balancing and thread pooling to achieve a high efficiency in the use of computer time and memory. The convergence of the variational and perturbative energies depends significantly on the orbitals used. The convergence obtained from using Hartree-Fock orbitals can be improved by using natural orbitals obtained from an SHCI calculation with a fairly large value of ϵ1, and can be further improved by using orbitals that minimize the SHCI variational energy using a modified version of the algorithm described in Ref. 121. We typically choose ϵ2 = 10−6ϵ1, so that a single parameter, ϵ1 controls the accuracy of the calculation. The energy at the ϵ1 = 0 limit is obtained using a quadratic fit to the energies versus the perturbative correction [106]. Note that although the SHCI algorithm has a perturbative component, systematically improvable approximations to the exact energy in the chosen basis are obtained by performing calculations with progressively smaller values of ϵ1 until the total energy (variational energy plus perturbative correction), or its extrapolation versus the perturbative correction, is converged to the desired tolerance. II. BASIS SETS, BOND LENGTHS, AND EFFECTIVE CORE POTENTIALS All calculations used the effective core potentials and associated aug-ccpVnZ gaussian basis sets of Trail and Needs[122–124]. These effective core potentials are produced from explicitly correlated multi-configuration Hartree- Fock calculations, and include contributions from core-core and core-valence correlations. Augmented double-, triple-, quadruple-, and quintuple-zeta basis sets were used. The molecules were computed with bond lengths in ˚A as follows: ScO: 1.668, TiO: 1.623,VO: 1.591, CrO: 1.621, MnO: 1.648, FeO: 1.616, CuO: 1.725. III. DATA Each contributing author provided a separate data file detailing the results of their calculations for both atomic and molecular systems. Each row of each data file indicates the system considered, the method and the basis set used, and the resulting total energies, along with any associated systematic or stochastic error. Additional method-specific information is also included (for example, the size of the active orbital space used in auxiliary-field quantum Monte Carlo). Not all methods completed each calculation using each level of basis set quality, due to the high computational expense of some methods. The CSV headers required are: • charge: charge of the molecule/atom (either 0 or 1) • molecule: (for molecules) the name of the molecule: example “VO” • atom: (for atoms) the name of the atom: example “V” • pseudopotential: always “trail” for this test set 10 • pyscf-version: “new” for after pyscf 1.5. A small improvement in accuracy was implemented after this version. Should be “new” for all calculations • method: a string representing the method used for the calculation. example “PBE” • totalenergy: total energy in Hartrees • totalenergy-stocherr : Stochastic error estimate • totalenergy-syserr : Systematic error estimate, if available A script called ’gather.py’ retrieves all data from all directories. Further scripts handle plotting and error estimation.", + "• method: a string representing the method used for the calculation. example “PBE” • totalenergy: total energy in Hartrees • totalenergy-stocherr : Stochastic error estimate • totalenergy-syserr : Systematic error estimate, if available A script called ’gather.py’ retrieves all data from all directories. Further scripts handle plotting and error estimation. IV. ENERGY COMPARISON FOR SEVERAL ACCURATE METHODS There are 5 methods (DMRG, iFCIQMC, UCCSD(T), AFQMC(MD) and SEET(FCI/GF2)) for which the total energies have an rms deviation of 4 mHa or less relative to the SHCI reference. The maximum absolute error, the rms error and the number of systems treated are shown in Table IV for these methods. Tables IV and IV show the corresponding quantities for the ionization energy and the dissociation energy. TABLE II. Total energy errors relative to SHCI of the five methods that agree best with SHCI. Method # systems max abs error rms error DMRG 39 0.001010 0.000238 iFCIQMC 49 0.001611 0.000639 UCCSD(T) 92 0.006811 0.002309 AFQMC(MD) 92 0.007470 0.003540 SEET(FCI/GF2) 59 0.013656 0.004001 TABLE III. Ionization energy errors relative to SHCI of the five methods that agree best with SHCI. Method # systems max abs error rms error DMRG 14 0.000826 0.000307 iFCIQMC 21 0.000965 0.000567 UCCSD(T) 28 0.001222 0.000675 AFQMC(MD) 28 0.008400 0.002888 SEET(FCI/GF2) 18 0.006580 0.002466 TABLE IV. Dissociation energy errors relative to SHCI of the five methods that agree best with SHCI. Method # systems max abs error rms error DMRG 7 0.000678 0.000327 iFCIQMC 1 0.000488 0.000488 UCCSD(T) 28 0.005108 0.002990 AFQMC(MD) 28 0.007880 0.002590 SEET(FCI/GF2) 14 0.008356 0.004152 11 0.15 0.20 Corr. en. (Ha) O 0.10 0.15 O+ 0.35 0.40 0.45 Sc 0.35 0.40 Sc+ 0.40 0.45 0.50 Ti 0.35 0.40 0.45 Corr. en. (Ha) Ti+ 0.45 0.50 0.55 V 0.4 0.5 V+ 0.5 0.6 Cr 0.45 0.50 0.55 Cr+ 0.5 0.6 Corr. en. (Ha) Mn 0.5 0.6 Mn+ 0.6 0.7 Fe 0.55 0.60 0.65 0.70 Fe+ 0.9 1.0 Cu 0.8 0.9 1.0 Corr. en. (Ha) Cu+ 0.9 1.0 MnO 0.75 0.80 0.85 TiO 0.00 0.05 1/n3 0.9 1.0 CrO 0.00 0.05 1/n3 0.8 0.9 VO 0.00 0.05 1/n3 1.1 1.2 1.3 Corr. en. (Ha) CuO 0.00 0.05 1/n3 1.0 1.1 FeO 0.00 0.05 1/n3 0.65 0.70 0.75 ScO FIG. 1. Basis set extrapolation of the SHCI correlation energy versus the n in the basis vnz. Only n values from 3-5 are shown since the n = 2 points deviate significantly from the straight lines shown. V. BASIS SET EXTRAPOLATION Methods not operating directly in the complete basis set limit must be extrapolated using finite basis calculations. The extrapolated energy Em(CBS) for method m is estimated from Em(CBS) = EHF(CBS) + ∆(CBS). (19) where EHF(CBS) is obtained by fitting the HF energies to the form EHF(n) = EHF(CBS) + b exp(−cn). (20) and ∆(CBS) is obtained by fitting the correlation energies to Em(n) −EHF(n) = ∆(CBS) + γ n3 , (21) where n is the cardinal index of the basis. An example extrapolation for all materials and SHCI correlation energies is shown in Fig 1. Extrapolations for the total energy, ionization energy", + "and ∆(CBS) is obtained by fitting the correlation energies to Em(n) −EHF(n) = ∆(CBS) + γ n3 , (21) where n is the cardinal index of the basis. An example extrapolation for all materials and SHCI correlation energies is shown in Fig 1. Extrapolations for the total energy, ionization energy and dissociation energy are shown in Tables V, VI and VII respectively. The ionization and dissociation energies converge much more rapidly than the total energies. Of the 4 basis set extrapolations considered, cbs45 is the most accurate. Table VIII shows the rms deviation of the total, ionization and binding energies of each basis set and extrapolation relative to the corresponding cbs45 energies. 12 TABLE V. SHCI total energies for all basis sets and extrapolations. cbs23 indicates an extrapolation using only vdz and vtz bases, and cbs34/cbs45/cbs345 are labeled in the same way. The differences between the cbs345 and the cbs45 extrapolations increase with the atomic number of the transition metal and range from 1-8 mHa. The cbs45 extrapolation is the most accurate one. Total Energy (Ha) basis O Sc Ti V Cr Mn Fe Cu vdz -15.781250 -46.396850 -57.883050 -71.077670 -86.612030 -103.942260 -123.518120 -197.236900 vtz -15.827020 -46.452300 -57.954630 -71.168980 -86.720920 -104.063660 -123.657200 -197.444640 vqz -15.839090 -46.474000 -57.983000 -71.205780 -86.767610 -104.115010 -123.716750 -197.536420 v5z -15.843170 -46.482130 -57.993860 -71.219830 -86.785320 -104.135420 -123.741800 -197.576080 cbs23 -15.846068 -46.475646 -57.985016 -71.207033 -86.766697 -104.114189 -123.715147 -197.531655 cbs34 -15.847828 -46.489701 -58.003293 -71.232489 -86.801544 -104.152337 -123.760024 -197.603262 cbs345 -15.847658 -46.490126 -58.004201 -71.233375 -86.802563 -104.154264 -123.763441 -197.609436 cbs45 -15.847432 -46.490691 -58.005411 -71.234555 -86.803920 -104.156833 -123.767993 -197.617661 Total Energy (Ha) basis O+ Sc+ Ti+ V+ Cr+ Mn+ Fe+ Cu+ vdz -15.300670 -46.156830 -57.634020 -70.828400 -86.372410 -103.673360 -123.233540 -196.967230 vtz -15.333580 -46.211490 -57.704240 -70.919550 -86.472980 -103.792140 -123.368570 -197.163040 vqz -15.342100 -46.233110 -57.732560 -70.957590 -86.518670 -103.842720 -123.427090 -197.253430 v5z -15.344790 -46.241150 -57.743160 -70.971820 -86.536270 -103.862840 -123.451830 -197.292510 cbs23 -15.348503 -46.234171 -57.733836 -70.957555 -86.515427 -103.840708 -123.424132 -197.245135 cbs34 -15.348554 -46.248705 -57.752710 -70.985120 -86.552131 -103.879380 -123.469470 -197.319302 cbs345 -15.348317 -46.248980 -57.753325 -70.985640 -86.553226 -103.881194 -123.472883 -197.325395 cbs45 -15.348001 -46.249346 -57.754146 -70.986333 -86.554685 -103.883611 -123.477431 -197.333514 Total Energy (Ha) basis ScO TiO VO CrO MnO FeO CuO vdz -62.420040 -73.903750 -87.085770 -102.558370 -119.850510 -139.435990 -213.123030 vtz -62.530130 -74.030850 -87.235260 -102.718670 -120.029280 -139.635460 -213.377780 vqz -62.568454 -74.075488 -87.288682 -102.779756 -120.097071 -139.711352 -213.484244 v5z -62.582141 -74.091473 -87.307394 -102.802737 -120.122364 -139.741682 -213.529128 cbs23 -62.576098 -74.084170 -87.297330 -102.786429 -120.103370 -139.717990 -213.483952 cbs34 -62.595849 -74.107270 -87.326938 -102.823579 -120.145888 -139.766095 -213.561561 cbs345 -62.596114 -74.107694 -87.326918 -102.824937 -120.147133 -139.769181 -213.567787 cbs45 -62.596466 -74.108258 -87.326891 -102.826745 -120.148791 -139.773293 -213.576082 TABLE VI. SHCI ionization energies. cbs23 indicates an extrapolation using only vdz and vtz bases, and cbs34/cbs45/cbs345 are labeled in the same way. Experimental values are also shown. Ionization Potential (Ha) basis Sc Ti V Cr Mn Fe Cu vdz 0.240020 0.249030 0.249270 0.239620 0.268900 0.284580 0.269670 vtz 0.240810 0.250390 0.249430 0.247940 0.271520 0.288630 0.281600 vqz 0.240890 0.250440 0.248190 0.248940 0.272290 0.289660 0.282990 v5z 0.240980 0.250700 0.248010 0.249050 0.272580 0.289970 0.283570 cbs23 0.241474 0.251180 0.249479 0.251269 0.273481 0.291015 0.286520 cbs34 0.240996 0.250583 0.247369 0.249413 0.272957 0.290555 0.283961 cbs345 0.241146 0.250876 0.247735 0.249337 0.273071 0.290558 0.284041 cbs45", + "0.269670 vtz 0.240810 0.250390 0.249430 0.247940 0.271520 0.288630 0.281600 vqz 0.240890 0.250440 0.248190 0.248940 0.272290 0.289660 0.282990 v5z 0.240980 0.250700 0.248010 0.249050 0.272580 0.289970 0.283570 cbs23 0.241474 0.251180 0.249479 0.251269 0.273481 0.291015 0.286520 cbs34 0.240996 0.250583 0.247369 0.249413 0.272957 0.290555 0.283961 cbs345 0.241146 0.250876 0.247735 0.249337 0.273071 0.290558 0.284041 cbs45 0.241346 0.251265 0.248222 0.249235 0.273222 0.290562 0.284147 exper 0.24113 0.25093 0.24792 0.24866 0.27320 0.29041 0.28394 13 TABLE VII. SHCI dissociation energies for the molecules considered in this work. cbs23 indicates an extrapolation using only vdz and vtz bases, and cbs34/cbs45/cbs345 are labeled in the same way. Dissociation Energy (Ha) basis ScO TiO VO CrO MnO FeO CuO vdz 0.241940 0.239450 0.226850 0.165090 0.127000 0.136620 0.104880 vtz 0.250810 0.249200 0.239260 0.170730 0.138600 0.151240 0.106120 vqz 0.255364 0.253398 0.243812 0.173056 0.142971 0.155512 0.108734 v5z 0.256841 0.254443 0.244394 0.174247 0.143774 0.156712 0.109878 cbs23 0.254384 0.253086 0.244229 0.173665 0.143113 0.156775 0.106230 cbs34 0.258321 0.256149 0.246622 0.174208 0.145724 0.158243 0.110471 cbs345 0.258330 0.255835 0.245885 0.174716 0.145210 0.158082 0.110693 cbs45 0.258343 0.255415 0.244903 0.175394 0.144526 0.157868 0.110989 TABLE VIII. RMS deviations of the SHCI total, ionization, and dissociation energies for various basis sets and extrapolations with respect to the cbs45 extrapolation. 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Trail and R. J. Needs, The Journal of Chemical Physics 146, 204107 (2017).", + "arXiv:nucl-th/0302049v1 19 Feb 2003 Parity-Violating Electron Scattering: How Strange a Future ∗ M.J. Ramsey-Musolf1, 2 1 California Institute of Technology, Pasadena, CA 91125 USA 2 Department of Physics, University of Connecticut, Storrs, CT 06269 USA I discuss several physics issues that can be addressed through the present and future program of parity-violating electron scattering measurements. In particu- lar, I focus on strange quark form factors, hadronic effects in electroweak radiative corrections, and physics beyond the Standard Model. PACS numbers: I. INTRODUCTION Parity-violating electron scattering (PVES) has become an established component of the nuclear physics research program at a variety of accelerators. Prior to the late 1990’s, the field was considered somewhat esoteric, with the completion of roughly one experiment per decade: the deep inelastic eD experiment at SLAC in the 1970’s[1]; the quasielastic beryllium experiment at Mainz in the 1980’s[2]; and the elastic carbon measurement culminating in a publication in 1990[3]. In the last few years, the rate at which new results are being reported has accelerated dramatically compared that of the past 30 years, and in the next several years, we anticipate even more results from the SAMPLE Collaboration at MIT-Bates[4], the Happex[5], G0[6], PAVEX[7], and Q-Weak[8] Collaborations at Jefferson Lab, the A4 experiment at Mainz[9], and the E158 experiment at SLAC[10]. Looking further down the road, the prospective up-grade of CEBAF may open up additional possibilities for PVES studies beyond those currently on the books. Suffice it to say, PVES has come into its own as an important tool for the study of a broad range of questions in nuclear and particle physics, moving far beyond its earlier incarnation as an almost exotic area of physics (for a review, see Ref. [11]). At the end of the day, what physics do we hope to have accomplished with this tool, and what important questions do we hope the field will have answered? In this talk, I ∗Talk given at PAVI02, workshop on parity violation in electron scattering, Mainz, Germany, June 2002. 2 would like to give my view. At the same time, I will discuss some recent developments that have opened up new directions for study of both hadron structure and particle physics. Of course, this theorist’s perspective will be weighted in favor of new theory, but I want to emphasize that these theoretical developments would have only academic interest were it not for the substantial progress on the experimental side. In fact, one of the attractive features of the PVES community is the close interplay between theory and experiment. The proper interpretation of the very tiny asymmetries measured with PVES requires careful theoretical delineation of various contributions. Conversely, the judicious choice of target and kinematics, together with hard-nosed experimental study of the systematic effects that could generate “false” asymmetries, provide theorists with a clean and unambiguous probe of nucleon structure and the weak interaction. This kind of syngergy between theory and experiment is a hallmark of our field. II. STRANGE QUARKS The present burst of activity in the field has been motivated by a desire to probe the strange quark", + "theorists with a clean and unambiguous probe of nucleon structure and the weak interaction. This kind of syngergy between theory and experiment is a hallmark of our field. II. STRANGE QUARKS The present burst of activity in the field has been motivated by a desire to probe the strange quark content of the nucleon. Understanding strange quarks is important from a number of perspectives: Flavor decomposition of nucleon properties. How much do the different flavors of light quarks contribute to the low-energy properties of the nucleon? For many years, the standard lore was that up- and down-quark degrees of freedom were sufficient to account for the most familiar aspects of the nucleon, such as its mass, spin, and magnetic moment. Analyses of the πN σ-term suggested that light-quarks were not, in fact, the whole story, and that possibly 20-30% of mN came from s¯s pairs[12]. Later, polarized deep inelastic scattering (DIS) measurements implied that only 30% of the nucleon spin came from quark spin, with the remainder being supplied by quark orbital angular momentum and by gluons[13]. Part of the picture associated with the “spin-crisis” was that the fraction of nucleon spin coming from s¯s pairs was about 10%, and that these pairs were polarized opposite to the direction of the total spin. In the quark model framework, this idea is a complete mystery. As a result, one would like to know what other nucleon properties are substantially affected by strangeness. Hence, the current program of measurements aimed at the strange quark contributions to nucleon electromagnetic properties. 3 The PVES strange quark program offers one advantage over other flavor decomposition studies, namely, the theoretically clean character of interpretation. Obtaining the s¯s con- tribution to the nucleon mass requires extrapolation of πN scattering amplitudes into the unphysical regime using a combination of dispersion relations and chiral perturbation theory (ChPT). The value for the ⟨N|¯ss|N⟩– relative to ⟨N|¯uu + ¯dd|N⟩– has varied somewhat over the years, in part due to the ambiguities associated with this extrapolation. The situation regarding the strange quark contribution to the nucleon spin, ∆s, is even more ambiguous. The extraction of ∆s from polarized DIS measurements relies on the as- sumption of flavor SU(3) symmetry. The relevant matrix element is not measured directly in the experiments, and information on neutron β-decay and hyperon semileptonic decays must be incorporated into the analysis using SU(3) symmetry. For some time, people have worried about the uncertainties associated with SU(3)-breaking. Recently, my collabora- tors and I studied these effects in ChPT and found them to be potentially serious[14, 15]. Specifically, one has ∆s = 3 2 [Γp + Γn] −5 √ 3 6 g8 A (1) = 0.14 −[0.12 + 0.25 + 0.10] , where the ΓN are the integrals of gN 1 (x) taken from experiment and g8 A is the eighth com- ponent of the octet of axial current matrix elements. A value for the latter can only be obtained from experimentally measured axial current matrix elements by invoking SU(3) symmetry. The chiral expansion of this SU(3) relationship is shown in the", + "taken from experiment and g8 A is the eighth com- ponent of the octet of axial current matrix elements. A value for the latter can only be obtained from experimentally measured axial current matrix elements by invoking SU(3) symmetry. The chiral expansion of this SU(3) relationship is shown in the square brackets in the second line of Eq. (1) (the first term is the experimental value for the sum of gN 1 integrals). The first, second, and third terms represent the O(p0), O(p2), and O(p3) con- tributions, respectively. It is a manifestly non-converging series. Based on the formula, we have no way of estimating the theoretical SU(3)-breaking uncertainty in ∆s, nor can we really say what the value of ∆s is. The advantage of PVES is that the interpretation of the results in terms of strange quarks does not suffer from these kinds of ambiguities. Once we have sufficiently precise measurements of the proton and neutron electromagnetic and weak neutral current form factors, we know the relative contributions of u, d, and s-quarks to the nucleon vector current matrix elements. The relative importance of the strange quark contribution may be either large or small, but either way, the answer will be clear. We will know the flavor 4 decomposition of the nucleon vector current matrix elements1. Sea quarks and the quark model. Why does the constituent quark model work so well in describing low-energy nucleon properties? In light of what we know about the presence of sea quarks and gluons in the nucleon, the success of the constituent quark description of the nucleon remains a puzzle. A variety of solutions have been proposed. It may be that sea quarks are simply “inert” when viewed over long-distance scales. A more interesting possibility is that the sea quarks are not inert but that their effects are hidden in the effective parameters of the quark model, such as the constituent quark mass and the “string tension” of the ¯qq potential2. It may also be that nature is more devious than either of these other possibilities suggest and has conspired to conceal the effects of sea quark dynamics through a series of cancellations. The latter view seems to emerge from dispserion relation analyses of isoscalar electromagnetic and strange quark form factors, as I discuss below. In any case, the beauty of the PVES measurements is that they probe the s¯s sea over distance scales relevant to the quark model description. If either the effective quark model parameters or nature itself has hidden sea quark effects, the measurements will lift the veil of secrecy and allow us to see what the sea is up to at low-energies. Z-Graphs vs. Loops. How do we relate the quark model description of the nucleon with one built out of hadronic degrees of freedom? In the latter framework, for example, the low-momentum, isovector electromagnetic properties of the nucleon involve a combination of the pion cloud and ρ-meson resonance. At the most naive level, one may think of the ρ-meson resonance as a “Z-graph” effect and the pion cloud as involving disconnected", + "degrees of freedom? In the latter framework, for example, the low-momentum, isovector electromagnetic properties of the nucleon involve a combination of the pion cloud and ρ-meson resonance. At the most naive level, one may think of the ρ-meson resonance as a “Z-graph” effect and the pion cloud as involving disconnected quark loops (see Fig. 1). In the t-channel, the ρ −γ coupling involves q¯q pair creation, and in order to connect to a nucleon, it must also involve the negative frequency part of the constituent quark propagator (the Z-graph). In the same channel, the pion cloud contains a ππ intermediate state. At the quark level, the non-resonant part of this contribution involves the presence of two q¯q pairs. With only three constituent quarks in the nucleon, this effect looks like a disconnected quark loop – a sea quark effect. On the other hand, in the non-relativistic quark model (NRQM), neither Z-graphs or loops are present, and yet 1 Kaplan and Manohar showed some time ago that the contributions from heavier quarks scale as (ΛQCD/MQ)4 and are, thus, negligible[16]. 2 I am indebted to Nathan Isgur for my understanding of this idea. 5 j\u0019 ih\u0019 j \u001a FIG. 1: Disconnected quark loop and Z-graph contributions to the nucleon isovector electromag- netic form factors. one can still reproduce the low-momentum isovector properties of the nucleon. How can this be? The strange quark contributions to the vector current form factors necessarily involve disconnected quark loops, since there are no valence strange quarks in the constituent quark model nucleon wavefunction. In a hadronic langauge, these loops look, for example, like K ¯K pairs. Dispersion relations also suggest that this kaon cloud is actually dominated by the φ resonance – a Z-graph effect that would imply more sizeable strangeness form factors than one gets from loops only. If this picture is confirmed by experiment, then it will force us to think more carefully about how to relate the quark and hadronic descriptions of the nucleon. Perturbation theory may not apply. The mass of the strange quark forces us into a theoretical no man’s land where we cannot easily apply our usual perturbative techniques for making predictions. The strange quark is not heavy enough to use heavy quark effective theory; since ΛQCD/ms ∼1, we have no small expansion parameter as we do for heavier quarks. At the same time, the strange quark may not be light enoughto make SU(3) chiral per- turbation theory applicable; mK/Λχ ∼1/2 – not a particularly small expansion parameter either. Thus, from the theoretical side, the study of strange quarks presents a particularly 6 challenging situation. Symmetry is impotent. Because the strange quark vector current contains a component that is a flavor SU(3) singlet, we cannot use SU(3) symmetry to relate the strangeness form factors to any other presently measured quantities. Specifically[17] ¯sγµs = JB µ −2JI=0 µ , (2) where JB µ is the baryon number, or SU(3) singlet current, and JI=0 µ is the isoscalar electro- magnetic (EM) current. We know quite a bit about the latter, but the PVES measurements will,", + "factors to any other presently measured quantities. Specifically[17] ¯sγµs = JB µ −2JI=0 µ , (2) where JB µ is the baryon number, or SU(3) singlet current, and JI=0 µ is the isoscalar electro- magnetic (EM) current. We know quite a bit about the latter, but the PVES measurements will, in fact, give us our first information on the form factors associated with JB µ . This means that, even if the SU(3) chiral expansion were well-behaved, there would appear coun- terterms or low-energy constants (LEC’s) that we cannot obtain from other experiments. Consequently, we cannot in general use ChPT to predict G(s) E and G(s) M , the strangeness electric and magnetic form factors, respectively. Clearly, the study of these form factors will lead to new insights into nucleon structure, and PVES provides us with a theoretically clean probe of this physics. The theoretical challenge is to translate the results of PVES measurements into a clear understanding of sea quark dynamics at distance scales for which traditional methods of treating nucleon structure run into trouble. A. Nonperturbative methods: recent developments and open questions Generally speaking, theorists have taken one of three approaches to the study of G(s) E and G(s) M : hadron models, dispersion relations, and lattice QCD. Of these, the latter two represent first principles methods at the hadronic and microscopic levels, respectively. Mod- els have their place, but they necessarily involve assumptions and approximations that are not always clearly identified or controlled. For this reason, we have witnessed a substantial number of model predictions for G(s) E and G(s) M and a correspondingly wide range of values. Should experiment agree with any subset of these calculations, it will be hard to gauge the significance of this agreement. For this reason, my bias favors the first principles treatments, even if they cannot yet be implemented in a completely successful way. Although – as I emphasized above – ChPT cannot provide predictions for G(s) E and G(s) M , it nevertheless provides a convenient framework in which to discuss the physics coming out 7 of other non-perturbative methods. For concreteness, I will focus on G(s) M at small q2. In this kinematic domain, one has an expansion G(s) M (q2) = µs + 1 6 < r2 s >M q2 + · · · . (3) The strangeness magnetic moment µs is given by[17, 18, 19] µs = 2MN Λχ ! bs + · · · (4) where bs is an a priori unknown low-energy constant that governs the lowest order [O(p2)] contribution and the + · · · denote higher order contributions arising from loops and sublead- ing operators. As noted earlier, one cannot use existing measurements to predict µs because bs contains an SU(3) singlet component. In contrast, the lowest order [O(p3)] contribution to the magnetic radius arises entirely from chiral loops[18, 19]: < r2 s >M= −πMN 3mKΛ2χ \u0010 5D2 −6DF + 9F 2\u0011 + · · · (5) where D and F are the usual SU(3) reduced matrix elements and the + · · · denote", + "the lowest order [O(p3)] contribution to the magnetic radius arises entirely from chiral loops[18, 19]: < r2 s >M= −πMN 3mKΛ2χ \u0010 5D2 −6DF + 9F 2\u0011 + · · · (5) where D and F are the usual SU(3) reduced matrix elements and the + · · · denote higher order contributions. Since D and F are known from other experiments, to O(p3) we have a parameter-free prediction for the strange magnetic radius. This result has been used to extrapolate the SAMPLE determination of G(s) M at Q2 = −q2 = 0.1 (GeV/c)2 to the photon point in order to obtain results for µs[4]. Given the size of the expansion parameter mK/Λχ, however, one might worry about the possible importance of higher-order contributions. To that end, the O(p4) contributions to < r2 s >M were recently computed in Ref. [19]. The loop effects arising at this order nearly cancel those appearing at O(p3), and one also encounters a new LEC. The resulting expression is < r2 s >M= −[0.04 + 0.3br s] fm2 , (6) where the first term in the square brackets gives the loop contribution and br s is the O(p4) magnetic radius LEC. On general grounds, one would expect the magnitude of br s to be of order unity with undetermined sign. In principle, then, its contribution could be an order of magnitude more important than the loop contribution. Until we are able to determine its value, we really do not know the either the magnitude or the sign of the slope of G(s) M at 8 the origin. As a result, the determination of G(s) M at a single kinematic point is not sufficient to determine µs. This new development has a couple of implications. First, in order to learn how much the strange quarks contribution to the nucleon magnetic moment, we will require an experimen- tal determination of the q2-dependence of G(s) M at low-q2. Accomplishing this determination will require completion of the the backward angle, low-q2 program as envisioned for the G0 experiment. Second, it means that understanding the non-perturbative dynamics of the strange sea – as parameterized by bs and br s – will be key to achieving a theoretical understanding of the strange magnetism. The strangeness electric form factor has a similar chiral expansion. Since the nucleon has no net strangeness, the leading q2-dependence is governed by the strangeness electric radius, < r2 s >E. In ChPT, this term arises at O(p3), and both loops and an electric radius LEC – cr s – appear at this order[17]: ⟨r2 s⟩E ≈[−0.15 + 0.17cr s] fm2 . (7) As in the case of the strangeness magnetic moment and radius, we require input from non- perturbative methods in order to obtain predictions for < r2 s >E. Lattice QCD. In principle, lattice QCD computations provide a means of obtaining values for the constants bs, br s, and cr s. The two most recent lattice computations provide conflicting results for these constants. Both computations were carried out in the quenched approxi- mation, so that s¯s pairs", + "s >E. Lattice QCD. In principle, lattice QCD computations provide a means of obtaining values for the constants bs, br s, and cr s. The two most recent lattice computations provide conflicting results for these constants. Both computations were carried out in the quenched approxi- mation, so that s¯s pairs appear only via the operator insertion. Even under this quenched approximation, computation of such “disconnected loop” contributions is difficult. For con- nected insertions, wherein the operator is inserted on a line originating from the hadronic source, one must compute quark propagators from a fixed point in spacetime to all possi- ble points on the lattice. In contrast, disconnected insertions require computation of quark propagators from all points on the lattice to all other points on the lattice. Carrying out such a computation is inordinately expensive. To get around this barrier, one employs “noise” methods, in which one computes quark propagators from a random selection of initial points to a random set of final spacetime points[20]. For a sufficiently large number of noise vectors, operator matrix elements computed using these quark propagators will approach the value one would obtain by computing propagators S(x, x′) for all x and x′. The two recent lattice computations adopted different philosophies for treating the noise 9 method. The Kentucky-Adelaide group employed a relatively small number of gauge field configurations but a large number of noise vectors[21]. They obtained statistically signficant signals for G(s) E and G(s) M at a variety of q2 values, and from these results, one may infer values for the strangeness LEC’s. In contrast, the computation of Ref. [22] used on the order of 1000 gauge field configurations but a relatively small number of noises. In this calculation, no significant signal was observed for the strangeness vector current matrix elements, but a non-zero value for the scalar density ⟨N|¯ss|N⟩was obtained. Resolution of this disagreement will require signficant future work, and the two groups are pursuing new, refined calculations of the strangeness form factors. Until a consensus emerges, it is difficult to draw any strong conclusions from lattice computations. For purposes of illustration, however, I will use the results from Ref. [21] below. Achieving firm lattice QCD predictions for ⟨N|¯sγµs|N⟩will ultimately require resolution of a number of issues. Clearly, one would like to have in hand an unquenched calculation. In addition, one would like to use sufficiently small values for the light quark masses that one can extrapolate to the physical values using ChPT. In this respect, use of chiral fermions – either domain wall or overlap – is likely to be critical. Recent work by the Kentucky group suggests that one must use very light quarks in order for the chiral extrapolation to be accurate, and in this respect, overlap fermions may provide some advantage[23]. Dispersion Relations. Athough the use of dispersion relations is not a microscopic method, it nevertheless provides a first principles approach to computing form factors in a hadronic basis. The inputs required include the analyticity properties of form factors, causality, unitarity, and experimental scattering amplitudes. To be concrete, the dispersion", + "advantage[23]. Dispersion Relations. Athough the use of dispersion relations is not a microscopic method, it nevertheless provides a first principles approach to computing form factors in a hadronic basis. The inputs required include the analyticity properties of form factors, causality, unitarity, and experimental scattering amplitudes. To be concrete, the dispersion relation for the strangeness magnetic and electric radii is < r2 s >M,E= 6 π Z ∞ 9m2π ImG(s) M,E(t) t2 dt . (8) The spectral function ImG(s) M,E(t) can be related to experimental scattering amplitudes by drawing upon some basic ideas in field theory. Symbolically, one has ImG(s) M,E(t) ∼PM,E X n < N ¯N|n >< n|¯sγµs|0 > , (9) where PM,E are magnetic and electric projection operators and the states |n⟩are all states carrying the quantum numbers of ¯sγµs. The lighest such state is the 3π state. Although 10 pions themselves have no valence strange quarks, three pions can resonate into an ω and, by virtue of ω-φ mixing, can give non-zero contributions to the strangeness vector current matrix elements. We have no indication that the 5π and 7π states resonate into an ¯ss vector meson, so they are usually omitted from the analysis3. In order of mass, the next state – and the most intuitive contributor – is the K+K−state. The ChPT expressions for loop contributions to µs and < r2 s >M,E are derived by computing the K+K−contribution at one-loop order. However, the work of Ref. [24] implies that the one-loop approximation is physically unrealistic. In order to satisfy the requirement of S- matrix unitarity, one must sum up kaon rescattering corrections to all orders. After doing so, one finds that the kaon cloud contribution to the spectral functions is dominated by the φ(1020) resonance. Moreover, the presence of this resonance signficantly enhances the kaon cloud contribution to the form factors. An interesting picture of form factor dynamics then emerges. To illustrate, consider the isoscalar and strangeness magnetic moments. In the case of µI=0, the 3π and K+K−con- tributes have nearly equal magnitude but opposite sign, and the resulting cancellation leads to the very small value for the isoscalar moment. For µs, in contrast, the 3π contribution gets suppressed by the ω-φ mixing angle (ǫ ∼0.05), while the kaon cloud contribution gets enhanced roughly by the inverse of the strange quark’s electric charge, or −3. The resulting value for µs is roughly −0.3 – sizeable enough to be seen by experiment. In short, nature conspires to hide the effects of resonating s¯s pairs in µI=0 through a cancellation against resonating light quark pairs. Far from being inert, the s¯s sea is quite active, though hidden from view in purely electromagnetic (EM) processes. Measuring the strangeness form factors directly should allow one to uncover the conspiracy. There is one possible loop hole in this picture. So far, we have only been able to include states up through the K+K−state in a rigorous way. The limitation is the absence of strong interaction scattering data involving higher-mass states. If the isovector EM form factors are any guide, then the low-mass states", + "possible loop hole in this picture. So far, we have only been able to include states up through the K+K−state in a rigorous way. The limitation is the absence of strong interaction scattering data involving higher-mass states. If the isovector EM form factors are any guide, then the low-mass states ought to suffice to describe the leading strangeness moments. Indeed, the non-resonant ππ continuum and ρ-meson resonance give nearly all of the low-q2 behavior of the isovector form factors. For the strangeness form factors, however, there is no guarantee that inclusion of the low-mass region is enough. Considerations of the 3 This approximation could entail introduction of some error, and it should be scrutinized further. 11 large q2-behavior of these form factors along with simple quark counting arguments suggests that higher-mass states (e.g., KKπ, etc.) may be required. Unfortunately, our ability to carry out a dispersion theory treatment of the higher-mass region is presently limited by the dearth of data. On the other hand, one could ultimately intepret the results of the G(s) E and G(s) M mea- surements in terms of the relative importance of this higher-mass region. If, for example, the value of µs differs substantially from −0.3 as suggested by the SAMPLE results, then there would have to be important higher-mass effects in order to overcome the fairly size- able low-mass contribution. Moreover, these higher-mass effects would have to be relatively strong for the strangeness current but relatively small for the isoscalar EM current. Such a situation would suggest that nature is an even more devious conspirator than we might otherwise think. Hadronic models. While I cannot review the full breadth of hadron model predictions for G(s) M,E, a few comments are warranted. From the standpoint of the dispersion theory treat- ment described above, many hadronic models are based on the idea of kaon cloud dominance. Intuitively, this idea is appealing. It says the strange sea gets polarized because the nucleon fluctuates into a kaon-hyperon pair. Many calculations – including some I carried out in the past – relied on one-loop computations to estimate this effect. The dispersion relation analysis, however, implies that the picture is physically unrealistic. Inclusion of rescattering effects, encoded by the LEC’s bs, br s, etc., is essential. Moreover, s¯s pair creation likely plays a more important role than the spatial separation of an s and ¯s. Finally, one may encounter cancellations between various contributions. In this respect, the quark model calculation of Geiger and Isgur is suggestive[25]. In that work, a sum over a tower of two hadron interme- diate states was considered, with the quark model providing the relevant couplings. As one proceeds up the tower, strong cancellations occur between various contributions, leading to small strangeness form factors in the end. By necessity, however, the calculation was carried out to one-loop level only. Whether this pattern of cancellations would persist after rescattering and resonance effects were included is hard to guess. Some Numbers. While the task of computing strange quark form factors has a long way to go, it is nonetheless useful to", + "however, the calculation was carried out to one-loop level only. Whether this pattern of cancellations would persist after rescattering and resonance effects were included is hard to guess. Some Numbers. While the task of computing strange quark form factors has a long way to go, it is nonetheless useful to discuss some of the numbers coming from the calculations. In Table I, I give some of the lattice QCD and dispersion relation values for the LEC’s bs, 12 Method bs + 0.6b8 br s cr s Lattice/Dipole Fit −0.6 ± 0.1 0.23 ± 0.13 −0.06 ± 0.4 Low-mass Dispersion Relation −0.6 −1.1 3.4 TABLE I: Low-energy constants for strange magnetic moment (bs), strange magnetic radius (br s), and strange electric radius (cr s). Lattice QCD values are obtained from Ref. [21]. “Low-mass” dispersion relation contributions obtained from Ref. [24]. br s, and cr s. These numbers should be taken with a grain of salt. As noted above, the two most recent lattice QCD computations differ substantially, with one seeing no signal for the strange vector current form factors. The G(s) M results of the Kentucky-Adelaide calculation were fit to a dipole form form, and from that a value for br s was obtained. There is no reason to believe the dipole form is correct, as an insufficient number of low-q2 values were computed to constrain the behavior of the form factor in this region. At the same time, the dispersion theory values result from only the low-mass part of the spectral function, and significant higher-mass contributions are not ruled out. With these caveats in mind, it is interesting to note that the results of the Kentucky- Adelaide lattice calculation of µs are consistent with the low-mass dispersion relation value, but rather substantial disagreement occurs for the magnetic and electric radii. Even the signs of the two calculations of < r2 s >M,E differ. Clearly, there is room for improvement in the theory. On the experimental side, completion of the entire program of PVES strange quark measurements is the only way to ensure that defensible theoretical progress is made. III. NEW WRINKLES: HADRON STRUCTURE While the current program of PVES measurements is far from complete, some intriguing puzzles have emerged from the results already reported, particularly in relation to elec- troweak radiative corrections. Theoretically, the challenge is to properly sort out the in- terplay between electroweak radiative corrections and strong interactions. In this context, a new idea for a future measurement of PV pion photoproduction on the ∆resonance has emerged recently – a measurement which may shed new light on an old problem in hadronic weak interactions. 13 A. Electroweak radiative corrections It is now well known that the importance of electroweak radiative corrections to the axial vector form factor are enhanced relative to the naively expected O(α/π) scale. This enhancement occurs because (a) the tree-level coupling is suppressed by the small vector coupling of the electron to the Z0, ge V = −1 + 4sin2 θW ≈−0.1, and (b) the corrections themselves contain large logarithms. These logarithms only appear in amplitudes involving", + "to the naively expected O(α/π) scale. This enhancement occurs because (a) the tree-level coupling is suppressed by the small vector coupling of the electron to the Z0, ge V = −1 + 4sin2 θW ≈−0.1, and (b) the corrections themselves contain large logarithms. These logarithms only appear in amplitudes involving γ-exchange – so they do not affect neutrino probes of the axial vector hadronic current. The large γ-exchange contributions can also be affected by parity-violating quark-quark interactions, which induce a parity-violating γNN coupling. People sometimes refer to this coupling as the nucleon anapole moment coupling. The anapole moment per se is not a physical observable, since its value depends on the choice of electroweak gauge parameter, ξ. Moreover, anapole moment effects can never be distinguished from other electroweak radiative corrections in any experiment. However, we might still use the “anapole moment” to refer to the ξ-independent contribution of the weak quark-quark interaction to the leading PV γNN coupling (for a discussion see Ref. [26]). It has been known for some time that because the anapole moment (AM) effects involve hadronic weak interactions, their magnitude is uncertain[27]. Thus, the theoretical predic- tion for Ge A – the axial vector form factor measured in PVES – has a fairly sizeable error bar. Were one to try and determine G(s) M from PV ep scattering alone, the theoretical Ge A uncertainty would add to the error bar for G(s) M . Hence, the SAMPLE collaboration has measured both the PV elastic ep and PV quasielastic eD asymmetries, which depend on different linear combinations of the two form factors in question and allow an experimental separation of Ge A(I = 1) from G(s) M . The theoretical uncertainty associated with Ge A(I = 0) has been estimated to be considerably smaller than for the isovector component, so the isoscalar contribution will be neglected in this discussion4. The published SAMPLE results[4] indicate that the radiative corrections to Ge A(I = 1) are, indeed, large and have the same sign as given by the calculations of Refs. [27, 28]. However, the magnitude may be considerably larger than predicted. If this result is confirmed by future measurements, such as the lower-Q2 SAMPLE deuterium experiment 4 Of course, the estimate of uncertainty for Ge A(I = 0) could be too small, a possibility that should be investigated. 14 or G0, one would like to understand what is behind the enhancement. Several ideas have been mentioned: Nuclear PV Effects. In the extraction of Ge A(I = 1) from the deuterium asymmetry, possible contributions to the PV γ-nucleus coupling from PV NN interactions have not previously been taken into account. In contrast to PV effects in elastic scattering, those entering inelastic scattering can give rise to a q2-independent contribution to the asymmetry: ALR(elastic) = A1q2 + · · · (10) ALR(inelastic) A0 + A1q2 + · · · . (11) The A1q2+· · · terms arise from the Z0-exchange amplitude and contain the usual electroweak form factors. The A0 term in ALR(inelastic) is generated by γ-exchange, where the γ couples to the", + "ALR(elastic) = A1q2 + · · · (10) ALR(inelastic) A0 + A1q2 + · · · . (11) The A1q2+· · · terms arise from the Z0-exchange amplitude and contain the usual electroweak form factors. The A0 term in ALR(inelastic) is generated by γ-exchange, where the γ couples to the nucleus through a PV transverse electric dipole matrix element. The presence of this term follows directly from an extension of Siegert’s theorem[29], which implies that for elastic scattering, all transverse electric dipole matrix elements are proportional to q2, while those entering inelastic transitions need not vanish at the photon point. In the case of A0, PV nucleon-nucleon interactions are responsible for the effect. People (myself included) have speculated that for sufficiently small |q2|, the A0 term may become quite important relative to the A1q2 term that contains Ge A, and that the larger-than-expected radiative corrections to Ge A may actually be a result of neglecting the A0 contribution. To address this question, two groups recently computed the q2-independent contribution generated by nuclear PV [30, 31]. The results of the two calculations are consistent with each other and indicate that the effect of nuclear PV is far too small to modify the value of Ge A extracted from the asymmetry. Generally speaking, one has[23] A0 A1q2 ∼10−4 m2 N |q2| . (12) At |q2| = 0.1 (GeV/c)2, the ratio is about 10−3. Thus, if the radiative corrections are truly enhanced, then some other mechanism would have to be the culprit. Box Graphs. One of the more sizeable contributions to the axial vector radiative corrections arises from the Z-γ box diagrams (Fig. 2a). The size of this amplitude is MγZ Box = Ve × h a0AI=0 N + a1AI=1 N i GF 2 √ 2 α 4π \u0014 ln MZ Λ + CA γZ(Λ) \u0015 , (13) 15 e p Z (a) e p Z (b) FIG. 2: (a) γZ box graph contribution to parity-violating ep scattering. (b) Zγ mixing tensor contribution to parity-violating ep scattering. where V µ e = ¯eγµe, AI=1 Nµ = ¯Nγµγ5τ3N, and AI=0 Nµ = ¯Nγµγ5N. The constants ai are given by a0 ≈1 6(3F −D)(9−20sin2 θW) and a1 = 1 2(D+F)(1−4sin2 θW). Note also the presence of the large logarithm. Here, Λ is a low-momentum cut-off, usually taken to be about one GeV, corresponding to the transition between the perturbative and non-perturbative domains. The constant CA γZ(Λ) encodes all the physics of the loop integral associated with momenta k < Λ. This physics includes hadronic form factors at the hadronic vertices as well as the full tower of hadronic intermediate states in the hadron propagator. To date, no one has attempted to carry out a systematic treatment of all of these effects. Only the nucleon intermediate state – along with the relevant form factors – has been included in model computations[27, 32]. One might wonder then, whether inclusion of other states, such as the uncorrelated πN, ∆, N∗, etc, might lead to an enhancement of this contribution. The problem is an open one and remains to be tackled.", + "along with the relevant form factors – has been included in model computations[27, 32]. One might wonder then, whether inclusion of other states, such as the uncorrelated πN, ∆, N∗, etc, might lead to an enhancement of this contribution. The problem is an open one and remains to be tackled. I like to emphasize that understanding this particular contribution could have implications for the interpretation for other precision electroweak measurements. For example, in neutron β-decay, the Wγ box graph produces a similar structure to that of Eq. (13). In this case, the A(lepton) ×V (N) amplitude is used to extract the value of Vud and test the unitarity of 16 the CKM matrix. A value for the constant CV γW(Λ) has been estimated, but as in the case of CA γZ(Λ), contributions from the full hadronic Green’s function have not been included. Were CA γZ(Λ) large enough to give the enhanced radiative corrections suggested by SAMPLE, and were CV γW(Λ) to have a similar magnitude, one would need to change the value of Vud substantially and engender serious problems with CKM unitarity. Although such a scenario seems unlikely, one might nevertheless gain new insights into the theoretical uncertainty associated with CV γW(Λ) through a study of CA γW(Λ). ¿From a somewhat different perspective, the SAMPLE Collaboration has also measured the parity-conserving asymmetry associated with transversely polarized electrons[33]. This asymmetry arises from the 2γ-exchange box graph, which is also sensitive to the low-energy part of the nucleon Green’s function. The experimental result for the asymmetry does not agree with theoretical predictions. Achieving a better understanding of the Zγ box graph may shed new light on this problem, and vice versa. Zγ Mixing. The largest contribution to the axial vector radiative corrections comes from the Zγ mixing tensor, ΠγZ (Fig. 2b). The light quark contribution to ΠγZ cannot be computed in perturbation theory, since the quarks have non-perturbative strong interac- tions. Traditionally, this problem is solved by relating the light quark component of ΠγZ to σ(e+e−→hadrons) using a dispersion relation and SU(3) arguments. For purely leptonic scattering, this approach ought to suffice. For eq scattering, however, one might worry about strong interactions between the quarks in ΠγZ and quarks in the proton. To some extent, these interactions have already been taken into account in anapole moment computations, but the matching of the two calculations onto each other remains to be worked out carefully. Again, it seems unlikely that some new effect might emerge when this matching has been achieved, but one should think about it nonetheless. B. PV ∆Electro- and Photoproduction: QCD Symmetries and Weak Interactions The G0 Collaboration plans a study of the q2-dependence of the axial vector N →∆ transition form factor, Ge∆ A . As in the case of Ge A, the transition axial vector form factor will be modified by electroweak radiative corrections. The same issues which apply to Ge A will also apply to Ge∆ A (for a detailed analysis, see Ref. [34]). In principle, these corrections could contain some q2-dependence not associated with the “primordial” form factor (as", + "the transition axial vector form factor will be modified by electroweak radiative corrections. The same issues which apply to Ge A will also apply to Ge∆ A (for a detailed analysis, see Ref. [34]). In principle, these corrections could contain some q2-dependence not associated with the “primordial” form factor (as one 17 might more directly measure with neutrino scattering). A more serious issue, however, arises because one is studying an inelastic process. For the same reasons as discussed above, the PV N →∆asymmetry will receive a q2-independent term as a consequence of Siegert’s theorem. In this case, the relevant E1 matrix element is parameterized by an LEC called d∆. At q2 = 0 one has[35] AN→∆ LR (q2 = 0) = −2 d∆ CV 3 mN Λχ , (14) where CV 3 is the leading vector current transition form factor. The presence of this term in the PVES asymmetry could complicate the determination of the q2-dependence of Ge∆ A , so it would be desirable to know it in some other way. In principle, a study of PV photoproduction could allow one to measure d∆, as discussed by JeffMartin elsewhere in these proceedings. A determination of d∆would also be interesting from another standpoint – namely, an old problem in nonleptonic weak interactions. In the ∆S = 1 sector, one has considerable knowledge of hyperon decays. The mesonic decays depend on both S- and P-wave ampli- tudes. To date, it has not been possible to obtain a successful, simultaneous description of both amplitudes in ChPT. Another problem appears with the ∆S = 1 radiative decays: B →B′γ. Here, one measures PV asymmetries αBB′ associated with the direction of the photon relative to the spin of the initially polarized hyperon. These asymmetries arise from the interference of an E1 and an M1 amplitude. In the limit of exact SU(3) flavor symmetry, the octet of baryons is degenerate and the E1 amplitude vanishes – a result known as Hara’s theorem[36]. In the real world, SU(3) symmetry is broken by the ms vs. mu + md mass dif- ference. Thus, one would expect the asymmetries to have a size αBB′ ∼ms/Λχ ∼0.15. The experimentally measured αBB′ are factors of four to five larger than this naive expectation. Based on general symmetry considerations, one has no way of explaining what is responsible for these enhancements. It would appear, then, that the application of QCD-based symmetries, such as chiral symmetry and SU(3) symmetry, to the theoretical analysis of ∆S = 1 non-leptonic weak interactions fails. One would like to know if this failure is a consequence of having strange quarks involved in the decay processes, or if it is a signature of a more fundamental feature of the interplay of strong and weak interactions. To that end, a study of d∆could provide new insight. The transition induced by the d∆operator is the ∆S = 0 analog of the ∆S = 1 E1 amplitudes governing the αBB′. If 18 N \u0001 3 2 \u0000H W eak N \u0001 1 2 \u0000FIG. 3: Resonance mixing contributions to parity-violating ∆photoproduction. the", + "study of d∆could provide new insight. The transition induced by the d∆operator is the ∆S = 0 analog of the ∆S = 1 E1 amplitudes governing the αBB′. If 18 N \u0001 3 2 \u0000H W eak N \u0001 1 2 \u0000FIG. 3: Resonance mixing contributions to parity-violating ∆photoproduction. the enhancements of the latter are not due to strange quarks but a more general feature of nonleptonic weak interactions, then one ought to observe an enhanced asymmetry for PV ∆photoproduction, which involves no strange quarks. On the other hand, if strangeness is the key to the breakdown of symmetry considerations, then the ∆S = 0 asymmetry ought to have its natural size. Recently, Holstein and Borasoy have developed a model that allows us to make this idea quantitative[37]. The idea is that the ∆S = 1 hyperon decays are affected by mixing of excited baryons into the ground states due to the hadronic weak interaction . In order to resolve the S-wave/P-wave problem, the mixing matrix elements WBB′(∆S = 1) must have a characteristic size larger than one might have naively expected. After determining what the WBB′(∆S = 1) must be to fix the S-wave/P-wave inconstency, Holstein and Borasoy used them to predict the αBB′ via resonance-mixing processes. Interestingly, the asymmetries are significantly enhanced by this mechanism, thereby helping to close the gap between the naive symmetry expectation and the experimental results. In principle, the same kind of weak interaction mixing could occur in the ∆S = 0 sector (see Fig. 3). If the mixing matrix elements WBB′(∆S = 0) have the same characteristic size as for the ∆S = 1 case, then one would expect d∆to be significantly enhanced – by 19 factors of 25 or more – compared to one’s naive expectation. We already know from studies of PV pp scattering that PV N →NM amplitudes (M is a meson) are of the same size as the ∆S = 1 B →B′π amplitudes, so it is not unreasonable to expect the WBB′ to have the same scale in the ∆S = 0 and ∆S = 1 sectors. In this case, AN→∆ LR (q2 = 0) would be rather large – on the order of ppm – and one could measure it with a reasonable amount of beam time. On the other hand, if one looked for an asymmetry of this size and didn’t see it, one might raise doubts about the Holstein/Borasoy proposal and, more generally, conclude that the problem with symmetry truly does hinge on having a strange quark involved. IV. NEW WRINKLES: THE STANDARD MODEL AND BEYOND The first few PVES measurements were intended to test the Standard Model (SM). Since then, the SM has been so well-tested in a variety of ways that we can, with a high degree of confidence, use the weak eq interaction as a well-understood probe of hadron and nuclear structure. The recent advances in this field – both experimental and theoretical – have now afforded us an opportunity to come full circle and again study fundamental aspects of the weak interaction. Of", + "high degree of confidence, use the weak eq interaction as a well-understood probe of hadron and nuclear structure. The recent advances in this field – both experimental and theoretical – have now afforded us an opportunity to come full circle and again study fundamental aspects of the weak interaction. Of course, there is little doubt that the SM is a successful model describing weak interac- tions far below the Planck scale. However, there exist many conceptual reasons for believing that the SM must be embedded in a larger theory. The quest to discover this “new” Stan- dard Model is at the forefront of both particle physics and nuclear physics. Unravelling the nature of the neutrino is clearly direction in which nuclear and particle physicists are pursuing this goal. Future collider experiments at the Large Hadron Collider and the Linear Collider will seek direct evidence for new physics. A third, and no less interesting, avenue is to carry out highly precise measurements of electroweak observables. Some such observables – such as the permanent electric dipole moments of the electron, neutron, and neutral atoms – are so small in the SM that the measurement of a nonzero effect would provide smoking gun evidence for new physics. Others, such as the PV asymmetries of interest here, have non-zero values in the SM, so one must attempt to probe for tiny deviations from the SM predictions. The presence or absence of such deviations – when taken in conjuction with other precision measurements – can provide important clues about the structure of the new SM. 20 To illustrate, recall how fits to precision electroweak observables predicted a range for the mass of the top quark prior to its discovery at the Tevatron. The subsequent discovery of a top quark with mass lying in this range provided beautiful confirmation of the SM at the level of quantum corrections. The same kind of synergy between indirect searches in precision electroweak measurements and direct collider searches will undoubtedly provide new insights about whatever lies beyond the SM. ¿From this standpoint, the quantity of interest to PVES is Qf W, the weak charge of a fermion or system of fermions. At tree level in the SM, it is the coupling of the Z-boson to the fermion f. It also governs the strength of the effective A(e) × V (f) interaction at Q2 = 0: Leff P V = −GF 2 √ 2Qf W ¯eγµγ5e ¯fγµf , (15) where, at tree-level in the SM, one has Qf W = 2If 3 −4Qfsin2 θW . (16) For both the electron and the proton, the magnitude of the weak charge is suppressed: Qp W = −Qe W ≈0.1. Because of this suppression, Qp,e W are rather transparent to new physics. As we have heard elsewhere in this conference, the goals of the PV M¨oller experiment at SLAC and the PV ep experiment at Jefferson Lab are to measure these two weak charges with some hope of finding evidence for new physics. We have also heard that even within the SM, these two measurements are", + "elsewhere in this conference, the goals of the PV M¨oller experiment at SLAC and the PV ep experiment at Jefferson Lab are to measure these two weak charges with some hope of finding evidence for new physics. We have also heard that even within the SM, these two measurements are quite interest- ing. The value of the effective weak mixing angle appearing in Eq. (16) actually depends on the scale at which one is carrying out the measurement. The SM predicts this scale- dependence. The SM prediction at the Z-pole has been confirmed with high precision, but there exist few precise determinations of sin2 θW(q2) below the Z0-pole. The cesium atomic PV experiment carried out by the Boulder group[38] probed the weak mixing angle – via the cesium nucleus weak charge (QCs W ) – at a very low scale. Although early indications sug- gested a substantial deviation from the SM, recent developments in atomic theory now imply the cesium atomic PV result is consistent with the SM. Since the extraction of QCs W from the measured PV transition rate depends on atomic structure calculations, there has always been some question about the theoretical uncertainty associated with this extraction. In the past couple of years, atomic theorists have computed several sub-one percent corrections – such as the Breit interaction correction[39], Uehling potential[40], and nucleus-enhanced 21 QED radiative corrections[41, 42]. As of this writing, there appears to be an emerging con- sensus that all of the important corrections have been properly taken into account and that the value of QCs W should be stable at the level of the experimental precision (about 0.4%). A value for sin2 θW(q2) at somewhat higher scales has been obtained by the NuTeV Collaboration using deep inelastic neutrino-nucleus scattering[43]. The result indicates a roughly 3σ deviation from the value predicted in the SM. As in the case of cesium APV, however, there exists some debate about the theoretical uncertainties associated with this extraction. It has been argued, for example, that small changes in parton distribution functions due to nuclear shadowing and isospin-breaking could substantially change the SM prediction for the neutrino scattering cross sections and reduce the significance of the NuTeV anomaly[44]. The Collaboration has responded to this suggestion with a vigorous rebuttal[45]. When the dust finally settles, one will have either learned something about parton distributions in nuclei or have striking evidence for new physics. In the context of these other measurements, the SLAC and JLab measurements offer several advantages. First, they are being undertaken at values of q2 lying between the cesium atomic PV and NuTeV scales. Second, they are both being performed at roughly the same scale: Q2 ≈0.03 (GeV/c)2. Third, they are theoretically clean, so their intepretation will not be subject to the kinds of controversies that have affected cesium atomic PV and the NuTeV result. This point is particularly important, so I will spend some time on it below. Finally, the two measurements are complementary. One is purely leptonic, while the other is semileptonic. As I will also discuss, a comparsion of the", + "of controversies that have affected cesium atomic PV and the NuTeV result. This point is particularly important, so I will spend some time on it below. Finally, the two measurements are complementary. One is purely leptonic, while the other is semileptonic. As I will also discuss, a comparsion of the two measurements can provide an interesting “diagnostic” tool for probing new physics. A. Theoretical Interpretability In principle, the PV M¨oller experiment, being purely leptonic, is the theoretically cleanest observable. There is some theoretical uncertainty in the SM prediction arising from the presence of light quark loops in ΠγZ5. This uncertainty is common to both Qe W and Qp W. Traditionally, the light quark contribution to ΠγZ is obtained by relating it to σ(e+e−→ hadrons) via a dispersion relation. In order to do so, one must also invoke SU(3) flavor 5 The issue raise earlier about interactions between these quarks and target quarks is less severe here. At q2 = 0, these effects are constrained by current conservation. 22 arguments. The uncertainty associated with this procedure was estimated some time ago by Marciano and Sirlin[32], and it falls below the expected experimental precision for both weak charge measurements. A second source of theoretical uncertainty arises from the extraction of Qe,p W from the measured asymmetries. In the case of the M¨oller experiment, one must be able to reliably subtract out contributions from the elastic and inelastic ep asymmetries. Although the number of ep events is small when compared with the number of M¨oller events, the inelastic ep asymmetries can be relatively large. The basic problem is that we do not know in detail what the inelastic asymmetries are. One strategy for circumventing it is to measure the total asymmetry in a kinematic region where the inelastic ep asymmetry is dominant, and then extrapolate to the kinematics of the M¨oller measurement. To the extent that various components of the inelastic asymmetry evolve with the relevant kinematic variables in a well understood way, this strategy should reduce the corresponding ep uncertainty to an acceptable level. In the case of Qp W, the design of the detector should eliminate any sizeable contribution from inelastic events. However, the elastic ep asymmetry itself receives hadronic contribu- tions in the guise of nucleon form factors. The associated uncertainty can also be reduced to an acceptable level by drawing on other measurements. Specifically, the asymmetry has the form Aep LR = GFQ2 4 √ 2πα h Qp W + F p(Q2, θ) i , (17) where F p(Q2, θ) is a function of nucleon form factors, both electromagnetic and strange. At forward angles and low-Q2, F p ∼Q2, yielding an overall Q4 contribution to the asymmetry. In contrast the Qp W term goes as Q2. The strategy, then, is to use the existing program of PV measurements to constrain the Q2-dependence of F p and extrapolate to very low- Q2 where the Q-Weak measurement will be performed and where the Qp W term gives the dominant contribution to the asymmetry. This strategy can be implemented in a rigorous way by", + "use the existing program of PV measurements to constrain the Q2-dependence of F p and extrapolate to very low- Q2 where the Q-Weak measurement will be performed and where the Qp W term gives the dominant contribution to the asymmetry. This strategy can be implemented in a rigorous way by using ChPT to express all the form factors in F p in terms of their known quark-mass dependence plus LEC’s. The existing program of experiments, plus the world data set for the electromagnetic form factors, will determine the LEC’s with sufficient precision to make the uncertainty in the low-Q2 asymmetry sufficiently small. No additional theoretical input on nucleon structure will be needed. Potentially more problematic considerations arise in arriving at the SM prediction for Qp W 23 itself. In particular, the two boson-exchange box graphs depend on hadronic physics via the nucleon Green’s function in the box. Fortunately, we are able to constrain the theoretical uncertainty associated with these effects to be acceptably small. The largest box graph contribution involves the exchange of two W-bosons. The total correction is about 26% of the tree-level value, largely due to the absence of the 1 −4sin2 θW factor from the MW W, the WW box amplitude. The latter is dominated by intermediate states of momentum k ∼MW, so the QCD corrections are perturbative. The latter can be computed using a combination of current algebra and the operator product expansion. The result is [46] MW W = −A(e) × V (p) GF 2 √ 2 ˆα 4πˆs2 \" 2 + 5 1 −αs(MW) π !# , (18) where ˆα is the fine structure constant evaluated at a scale µ = MW in the MS scheme and ˆs2 is the corresponding definition of sin2 θW. Inclusion of the O(αs) contributions generate a ∼−0.7% correction to Qp W, and higher-order corrections in αs will be at roughly an order of magnitude less important. Note that the QCD corrections involve no large logarithms, since the currents involved are conserved (or partially conserved). A similar correction applies to the ZZ box graphs, but the effect in this case is much smaller – roughly −0.1% in Qp W. In contrast, the γZ box graphs receive contributions from both low- and high-momentum scales. This sensitivity to the different scales is reflected in the presence of a large logarithm in the amplitude: MγZ = −A(e) × V (p) GF 2 √ 2 5ˆα 2π(1 −4ˆs2) \" ln M2 Z Λ2 ! + CV γZ(Λ) # , (19) where I have omitted contributions suppressed by powers of pe/MZ. As in Eq. (13), Λ is a scale associated with the transition from the non-perturbative to the perturbative region and is on the order of 1 GeV. At present, we cannot reliably compute contributions to the loop integral arising from momenta below this scale. Instead, we parameterize their effects by the constant CV γZ(Λ). Fortunately, the effect of this unknown term is suppressed by the overall 1 −4ˆs2 prefactor, so we can live with a fairly large uncertainty in CγZ without affecting the interpretation of", + "the loop integral arising from momenta below this scale. Instead, we parameterize their effects by the constant CV γZ(Λ). Fortunately, the effect of this unknown term is suppressed by the overall 1 −4ˆs2 prefactor, so we can live with a fairly large uncertainty in CγZ without affecting the interpretation of Qp W. The presence of this prefactor is, itself, something of a fortunate accident. It arises from the sum of the box and crossed-box graphs. The spacetime structure of this sum dictates that the A(e)×V (p) amplitude arises from the antisymmetric product of the electron electromagnetic current and the vector part of the electron weak 24 neutral current. The latter contains the 1 −4ˆs2 vector coupling of the electron to the Z0, thereby producing the suppression factor in MγZ. To estimate the uncertainty associated with CV γZ we can follow a philosophy adopted by Sirlin in his classic analysis of neutron β-decay[47]. In that case, a similar unknown constant arises from the γW box graph. The size and uncertainty of CV γW is constrained because the vector part of the decay amplitude is used to determine Vud6. In order for Vud to be consistent with the unitarity of the CKM matrix, the uncertainty from CV γW cannot be larger than about ±2. The hadronic dynamics responsible for CV γW and CV γZ are similar, so one would expect the uncertainty in the two to be comparable. The corresponding uncertainty in Qp W is about ±1.5%. A siginficantly larger uncertainty in CV γW and CV γZ would imply unacceptably large deviations from CKM unitarity. Of course, one should pursue computations of these quantities from first principles – possibly using lattice QCD. However, the fortuitous presence of the 1 −4ˆs2 suppression factor has given us a sizeable zone of theoretical safety. One other hadronic effect which one might worry about for Qp W is the impact of isospin admixtures into the proton wavefunction. On general grounds, one would expect isospin- breaking effects to be of O(α) and, therefore, a source of concern. At Q2 = 0, however, a diagonal version of the Ademollo-Gatto theorem[48] implies that isospin-breaking effects in the matrix element of the hadronic vector current vanish identically[46]. They only turn on for non-zero q2. In this case, they will be effectively contained in the measured form factor term F p and constrained – along with all other q2-dependent effects – by experiment. B. New Physics Given the theoretical interpretability of the two weak charge measurements, one can credibly view them as probes physics beyond the SM. I have been thinking quite a bit about supersymmetry (SUSY) recently, so I will focus my discussion on this brand of new physics. SUSY remains one of the most strongly-motivated extensions of the Standard Model for a number of reasons: it gives a solution to the “hierarchy problem” in particle physics (essentially, the question as to why the electroweak scale is stable); it generates unification 6 This amplitude is actually known most precisely from superallowed Fermi nuclear β-decay, though new experiments using ultracold neutrons should provide", + "a number of reasons: it gives a solution to the “hierarchy problem” in particle physics (essentially, the question as to why the electroweak scale is stable); it generates unification 6 This amplitude is actually known most precisely from superallowed Fermi nuclear β-decay, though new experiments using ultracold neutrons should provide competitive determinations. 25 of the electroweak and strong couplings close to the Planck scale; it gives a natural candidate for cold dark matter (the neutralino, ˜χ0); and it contains new sources of CP violation that could help us account for the observed predominance of matter over anti-matter (the baryon asymmetry of the universe). Of course, since no one has yet seen any of the “superpartners” of SM particles, they must be heavier than the particles in the SM. Consequently, SUSY must be a broken symmetry. The breaking cannot be too large, however; otherwise, the hierarchy problem re-emerges. Future collider experiments at the LHC and LC will go looking for the heavy superpart- ners. One might also ask about observing their effects indirectly. For example, superpartners can contribute to precision electroweak observables through radiative corrections. Perhaps, the most well-known example of such effects are superpartner loop contributions to the muon anomalous magnetic moment. On general grounds, one expects the size of SUSY radiative corrections to go as α/π(M/ ˜ M)2, where M is a SM mass and ˜ M is a superpartner mass. For the muon g −2, one has M = mµ. Taking ˜ M ∼100 GeV, one gets an effect of the order of 10−9 – roughly the size of the current experimental error bar. The possible significant deviation of (g −2)µ from the SM prediction would be consistent with SUSY loop contribu- tions containing superpartners of mass of a few hundred GeV. For this reason, the particle physics community has become quite excited by new results for the muon anomaly7. For weak interaction processes, such as β-decay and PVES, one has M = MW. Conse- quently, the magnitude of SUSY loop effects, relative to the SM prediction, is generically of order 10−3. However, since Qe,p W are suppressed in the SM, one actually needs a precision of order a few percent – rather than a few tenths of a percent – to be sensitive to SUSY radiative corrections. For this reason, it is interesting to study the effect of these radiative corrections on the weak charges. To that end, we need to modify our formula for Qf W as follows: Qf W = ρP V \u0010 2If 3 −4QfκP V ˆs2\u0011 + λf . (20) Here, ρP V and κP V are universal, in that they do not depend on the species of fermion being probed with PVES. The quantity λf, in contrast, is species-dependent. At tree-level, one has ρP V = 1 = κP V and λf = 0. Inclusion of loops lead to deviations from these values. In 7 There remains some controversy over the theoretical uncertainty in the SM prediction associated with hadronic contributions. I will not discuss this controversy here. 26 particular, one can think of", + "V = 1 = κP V and λf = 0. Inclusion of loops lead to deviations from these values. In 7 There remains some controversy over the theoretical uncertainty in the SM prediction associated with hadronic contributions. I will not discuss this controversy here. 26 particular, one can think of δκP V as the change in the apparent weak mixing angle due to loop effects. The effects of new physics on κP V arise in two ways: first, one has a change in the prediction for ˆs2 due to its defintion in terms of other more precisely known quantities (α, Gµ, and MZ). This definition gets modified by new physics. Second, δκP V receives contributions directly from the PV ef amplitudes. In order to analyze the corrections δρsusy P V , δκsusy P V , and λsusy f one needs to compute a large number of loop graphs. Moreover, one has to take into account the effects of SUSY- breaking. In general, this task is non-trivial. In the Minimal Supersymmetric Standard Model (MSSM), SUSY-breaking is governed by 105 parameters. To simplify the problem – and to build a real theory – people have invented models of SUSY-breaking, such as supergravity, which reduce the number of independent parameters to a handful. Whether or not these model assumptions are entirely consistent with electroweak data is not entirely clear. We recently completed an analysis of charged current observables and found that the superpartner spectrum implied by SUSY-breaking models may not be consistent with the precision data[49]. For this reason, it is desirable to carry out a model-independent analysis of SUSY radiative corrections, thereby avoiding the simplifying assumptions of SUSY-breaking models. My collaborators and I have completed such a model-independent analysis of SUSY radiative corrections to Qe,p W in the MSSM[50]. We found that the effects on the two weak charges are highly correlated, and that the relative sign of the correction in both cases is positive over nearly all of the available SUSY-breaking parameter space. This correlation – shown in Fig. 4 – arises because the dominant effect occurs via δκsusy P V , with some scatter due to the effects of δρsusy P V and λsusy f . Moreover, the potential size of the SUSY radiative corrections could be as large as the projected experimental error bars. What this means is that if both experiments come in consistent with the SM, then one could not say much about the SUSY-breaking parameters. However, deviations from the SM of the order of 2σ or more would – depending on the relative sign of the effect – start to either favor or exclude parts the parameter space. The situation is helped somewhat by the correlation between the two weak charges in SUSY. The two measurements together really act like one determination of δκsusy P V with better statistics than obtained with either alone. Nevertheless, it would be advantageous –looking further down the road – to pursue even more precise measurements of the weak charges. 27 FIG. 4: Relative shifts in the electron and proton weak charges due", + "act like one determination of δκsusy P V with better statistics than obtained with either alone. Nevertheless, it would be advantageous –looking further down the road – to pursue even more precise measurements of the weak charges. 27 FIG. 4: Relative shifts in the electron and proton weak charges due to supersymmetric effects. Dots indicate loop contributions, where each point corresponds to a randomly-selected set of soft SUSY- breaking parameters. Interior of truncated ellipse denotes possible shifts from R parity-violating interactions. The effects of SUSY can also arise at tree-level if one allows for the violation of lepton number (L) and/or baryon number (B). If B-L is conserved, then the MSSM has another symmetry called “R parity”. Conservation of R parity is not automatic; it has to be imposed by hand. There are some phenomenological constraints. The absence of proton decay highly constrains the possibility of B-violating interactions. However, the constraints on L-violating sources of R-parity violation (RPV) are much less severe. The existence of any such interactions would, however, sound the death knell for SUSY dark matter, since the lightest superpartner (presumably the ˜χ0) would no longer be stable. Thus, one would like to know from experiment whether or not the MSSM respects R parity. To that end, a comparison of Qe W and Qp W measurements could be quite interesting. In Fig. 4, I show the possible deviations of these quantities from the SM values due to RPV effects. The interior of the truncated ellipse is the region allowed by other precision measurements at 95% confidence. Note that for most of this region, the relative sign of the deviation in Qe W and Qp W is negative – in contrast to the situation for SUSY loops. 28 In fact, there is nearly no overlap between the regions in this plane associated with the different effects. Thus, a comparison of these two measurements could, in principle, serve as a “diagnostic” for SUSY with our without R parity. In fact, the diagnostic potential of these two experiments is even greater if one considers other models of new physics. For example, E6 grand unified theories allow for the possibility of a relatively low mass, extra neutral gauge boson8. In these models, the effect of a Z′ on Qe W and Qe W would also be correlated (having the same relative sign). However, a sizeable shift in these weak charges would also imply a sizeable deviation of the cesium weak charge from the SM prediction. Since QCs W currently agrees with the SM, this scenario appears to be disfavored. In contrast, the effects of SUSY loops on QCs W is tiny – below the current experimental plus theoretical error – due to cancellations between up- and down-quark contributions. Thus, one could still have sizeable loop-induced shifts in Qe,p W without being inconsistent with cesium atomic PV. Similarly, the allowed RPV region shown in Fig. 4 already incorporates the constraints from the QCs W determination. V. CONCLUSIONS I am convinced that PVES will remain a highly interesting field for some time to come. In addition", + "loop-induced shifts in Qe,p W without being inconsistent with cesium atomic PV. Similarly, the allowed RPV region shown in Fig. 4 already incorporates the constraints from the QCs W determination. V. CONCLUSIONS I am convinced that PVES will remain a highly interesting field for some time to come. In addition to providing a definitive answer to the question about strangeness, PVES has also opened up possibilities for studying other aspects of hadronic and nuclear structure9. These studies rely on our present understanding of the weak neutral current interaction in order to probe novel aspects of low-energy QCD. At the same time, the success of this program has opened the way for new experiments to study the weak interaction itself. Looking further into the future, one might anticipate new PV deep inelastic scattering experiments after the CEBAF upgrade that may offer new glimpses of higher-twist physics, or even a more precise measurement of the M¨oller asymmetry10. Suffice it to say, however, that completion of the current program of PVES measurements is essential to the long-term success of the field. ACKNOWLEDGEMENTS 8 The E6 models are actually a very broad class of models, though they do not encompass all possibilities for a “low-mass” Z′. 9 In the interest of time, I have not discussed the important measurement of the neutron distribution in lead [7]. 10 I am indebted to Paul Souder, Paul Reimer, and Dave Mack for a discussion of these possibilities. 29 I would like to thank the organizers of this workshop and the support stafffor their hospitality during my stay in Mainz. I also thank A. Kurylov for assistance in preparing this manuscript. 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Su, [hep-ph/0205183] (2002).", + "arXiv:1009.5232v2 [physics.gen-ph] 12 Aug 2011 Noname manuscript No. (will be inserted by the editor) Peter Leifer The quantum origin of inertia and the radiation reaction of self-interacting electron Received: date / Accepted: date Abstract The internal structure of self-interacting quantum particle like electron is independent on space-time position. Then at least infinitesimal kinematic space-time shift, rotation or boost lead to the equivalent inter- nal quantum state. This assumption may be treated as internal (quantum) formulation of the inertia principle. Dynamical transformation of quantum setup generally leads to deformation of internal quantum state and measure of this deformation may be used as quantum counterpart of force instead of a macroscopic acceleration. The reason of inertia arises, thereby, as a conse- quence an internal motion of quantum state and its reaction on dynamical quantum setup deformation. The quantum origin of the inertia has been discussed in this article in the framework of “eigen-dynamics” of self-interacting relativistic extended quan- tum electron. Namely, a back reaction of spin and charge “fast” degrees of freedom on “slow” driving environment during “virtual measurement” leads to the appearance of state dependent non-Abelian gauge fields in dynami- cal 4D space-time. Analysis of simplified dynamics has been applied to the energy-momentum behavior in the relation with runaway solutions of previ- ous models of an electron. 1 Introduction. An “internal” formulation of the inertia principle The fundamental question of the nature of inertial mass is not solved up to now. Success of Newton’s conception of physical force influencing on a Peter Leifer Haatid, College for Science and Technology, 18123 Afula, Israel Tel.: +972-4-6405580 Fax: +972-4-6405576 E-mail: leifer@bezeqint.net 2 separated body may be explained by the fact that the geometric counterpart to the force F - acceleration a in some inertial frame was found with the simplest relation a = F m to the mass m of a body. The consistent formulation of mechanical laws has been realized in Galilean inertial systems. The class of the inertial systems contains (by a convention) the one unique inertial system - the system of remote stars and any reference frame moving with constant velocity relative these remote stars. Then, on the abstract mathe- matical level arose a “space” - the linear Euclidean space with appropriate vector operations on forces, momenta, velocities, etc. General relativity and new astronomical observations concerning accelerated expansion of Universe show that all these constructions are only a good approximation, at best. The line of Galileo-Newton-Mach and Einstein (with serious reservations about conception of the “space”) argumentations made accent on some abso- lute global reference frame associated with the system of remote stars. This point of view looks as absolutely necessary for the classical formulation of the inertial principle itself. This fundamental principle has been formulated, say, “externally”, i.e. as if one looks on some massive body perfectly isolated from rest Universe. In such approach only “mechanical” state of relative mo- tion of bodies expressed by their coordinates in space has been taken into account. Nevertheless, Newton clearly saw some weakness of such approach. His famous example of rotating bucket with water shows", + "on some massive body perfectly isolated from rest Universe. In such approach only “mechanical” state of relative mo- tion of bodies expressed by their coordinates in space has been taken into account. Nevertheless, Newton clearly saw some weakness of such approach. His famous example of rotating bucket with water shows that there is an absolute motion since the water takes on a concave shape in any reference frame. Here we are very close to different - “internal” formulation of the iner- tia principle and, probably, to understanding the quantum nature of inertial mass. Namely, the “absolute motion” of a body should be turned towards not outward, to distant stars, but inward – to the deformation of the body. This means that external force not only changes the inertial character of its motion: body accelerates, moreover – the body deforms. Two aspects of a force action: acceleration relative inertial reference frame and deformation of the body are very important already on the classical level as it has been shown by Newton’s bucket rotation. The second aspect is es- pecially important for quantum “particles” since the acceleration requires the point-like localization in space-time; such localization is, however, very problematic in quantum theory. Nevertheless, almost all discussions in foun- dations of quantum theory presume that space-time structure is close with an acceptable accuracy to the Minkowski geometry and may be used with- out changes in quantum theory up to Planck’s scale or up to topologically different space-time geometry of string theories. Under such approach, one loses the fact that space-time relationships and geometry for quantum ob- jects should be reformulated totally at any space-time distance since from the quantum point of view such fundamental dynamical variables as “time- of-arrival” of Aharonov et al. (1998) and a position operator of Foldy & Wouthuysen (F.-W.) (1950) and Newton & Wigner (1949) representations are state-dependent (Leifer, 1997,1998,2004,2005,2007,2009,2010). Therefore space-time itself should be built in the frameworks of a new “quantum ge- ometry”. In such a situation, one should make accent on the second aspect of the force action – the body deformation. In fact, microscopically, it is already 3 a different body with different temperature, etc., since the state of body is changed (Leifer, 1988). In the case of inertial motion one has the opposite situation – the internal state of the body does not change, i.e. body is self- identical during inertial space-time motion. In fact this is the basis of all classical physics. Generally, space-time localization being treated as ability of coordinate description of an object in classical relativity closely connected with operational identification of “events” (see (Einstein, 1905)). It is tacitly assumed that all classical objects (frequently represented by material points) are self-identical and they cannot disappear during inertial motion because of the energy-momentum conservation law. The inertia law of Galileo-Newton ascertains this self-conservation “externally”. But objectively this means that physical state of body (temporary in somewhat indefinite sense) does not de- pend on the choice of the inertial reference frame. One may accept this state- ment as an “internal” formulation of the inertial law", + "law. The inertia law of Galileo-Newton ascertains this self-conservation “externally”. But objectively this means that physical state of body (temporary in somewhat indefinite sense) does not de- pend on the choice of the inertial reference frame. One may accept this state- ment as an “internal” formulation of the inertial law that should be of course formulated mathematically. I put here some plausible reasonings leading to such a formulation. Up to now the localization problem of quantum systems in the space-time is connected in fact with the fundamental classical notion of potential energy and force. Einstein and Schr¨odinger already discussed the inconsistency of usage such purely classical notions together with the quantum law of motion and the concepts of “particle” and “acceleration” as well (see one of the letter of Einstein to Schr¨odinger (Einstein, 1950), and the article (Einstein, 1953)). But these messages are left almost without attention by the physical community. Now one should pay the bill. Newton’s force is the physical reason for the absolute change of the char- acter of a body motion expressed by acceleration serves as geometric coun- terpart to the force (curvature of the world line in Newtonian space and time is now non-zero). However there is no adequate geometric notion in quan- tum theory since, for example, the notion of trajectory of quantum system in space-time was systematically banned. The energy of interaction expressed by a Hamiltonian Hint is in some sense an analoge of a classical force. Gener- ally, this interaction leads to the absolute change (deformation) of the quan- tum state (Leifer, 1998). Notice, quantum state is in fact the state of motion (Dirac,1928). Such motion takes the place in a state space modeled frequently by some Hilbert space H. But there is no geometric counterpart to Hint in such functional space. In order to establish the geometric counterpart to Hint it is useful initially to clarify the important question: what is the quantum content of a classical force, if any? Let me use a small droplet of mercury as simple example of macroscopic system. The free droplet of mercury is in the state of the inertial motion “whether it be of rest, or of moving uniformly forward in a straight line”. In fact this statement means that physical states of the droplet (its internal de- grees of freedom) being in inertial motion are physically non-distinguishable. The force applied to the droplet breaks the inertial character of its motion and deforms its surface, changes its surface tension temperature, etc. In fact an external force perturbs Goldstone’s modes supporting the droplet as a macro-system (Umezawa, Matsumoto & Tachiki, 1982) and micro-potentials acting on any internal quantum particle, say, electrons inside of the droplet. It 4 means that quantum states and their deformations may serve as a “detector” of the “external force” action on the droplet. Therefore it is reasonable to use quantum state deformations as a counter- part instead of classical acceleration, since generally acceleration depends on mass, charge, etc., that is impossible to establish a pure space-time geomet- rically invariant counterpart of a classical", + "a “detector” of the “external force” action on the droplet. Therefore it is reasonable to use quantum state deformations as a counter- part instead of classical acceleration, since generally acceleration depends on mass, charge, etc., that is impossible to establish a pure space-time geomet- rically invariant counterpart of a classical force independent on a material body. Only a classical gravitation force may be geometrized assuming the gravitation field may be replaced locally by an accelerated reference frame since in the general relativity (GR) the gravitation field and accelerated refer- ence frame are locally non-distinguishable. There is, however, a more serious reason why space-time acceleration cannot serves as the completely robust geometric counterpart of the force. The physical state of the droplet freely falling in the gravitation field of a star is non-distinguishable from the physical state of the droplet in an remote from stars area. It means that from the point of view of the “physical state” of the droplet, the class of the inertial systems may be supplemented by a reference frame freely falling in a gravitation field. Therefore macroscopic space-time acceleration cannot serve as a discriminator of physical state of body. Thus, instead of choosing, say, the system of distant stars as an “outer” absolute reference frame (Einstein, 1916) the deformation of quantum state of some particle of the droplet may be used. It means that the deformation of quantum motion in quantum state space serves as an “internal detector” for “accelerated” space-time motion. This deformation being discussed from the quantum point of view gives the alternative way for the connection of Hint action with a new geometric counterpart of interaction - coset structure of the quantum state space. It means that instead of absolute external reference frame of remote stars one may use “internal”, in fact a quantum reference frame (Leifer, 2010,2009,1997,2007,2004,1998,2005). There is different description of accelerated electron in the thermalized vacuum (Unruh, 1976), (Bell & Leinaas, 1983), (Bell& Leinaas, 1986), (Myhrvold, 1985). Avoiding detailed analysis I should note that any variant of such de- scription uses the interaction of “detector” with quantum field embedded in single space-time where the acceleration has absolute sense. Indeed, Unruh temperature TU = ℏa 2πckB is proportional to the acceleration a. Relative what reference frame? Say, freely falling electron is definitely “accelerated” relative remote stars, but its quantum state should be the same as the quantum state of “free” electron in remote from masses area. Therefore, I have assumed that the equivalence of pure quantum state of single free and freely falling electron (Leifer, 2009). Then the deformations of quantum motion generated by the coset action in the quantum state space will be used as an invariant coun- terpart of a “quantum force” applied to self-interacting electron in dynamical space-time. It means nothing but in the developing theory a distance between quantum states in the state space should replace a distance between “bod- ies” in space-time as the primary geometric notion. Thus, the mathematical formulation of the quantum inertia law requires the intrinsic unification of relativity and quantum principles. 5 2 Intrinsic", + "nothing but in the developing theory a distance between quantum states in the state space should replace a distance between “bod- ies” in space-time as the primary geometric notion. Thus, the mathematical formulation of the quantum inertia law requires the intrinsic unification of relativity and quantum principles. 5 2 Intrinsic unification of relativity and quantum principles The mentioned above the localization problem in space-time and connected with this deep difficulties of divergences in quantum field theory (QFT) insist to find a new primordial quantum element instead of the classical “material point”. Such element is the quantum motion - the quantum state of a system (Dirac, 1928). Quantum states of single quantum particles may be repre- sented by vectors |Ψ >, |Φ >, ... of linear functional Hilbert space H with finite or countable number of dimensions. It is very important to note that the correspondence between quantum state and its vectors representation in H is not isomorphic. It is rather homomorphic, when a full equivalence class of proportional vectors, so-called rays {Ψ} = z|Ψ >, where z ∈C \\ {0} corre- sponds to one quantum state. The rays of quantum states may be represented by points of complex projective Hilbert space CP ∞or its finite dimension subspace CP(N −1). Points of CP(N −1) represent generalized coherent states (GCS) that will be used thereafter as fundamental physical concept instead of “material point”. This space will be treated as the space of “un- located quantum states” in the analog of the “space of unlocated shapes” of (Shapere & Wilczek, 1989). The problem we will dealing with is the lift the quantum dynamics from CP(N −1) into the space of located quantum states. That is, the difference with the Shapere & Wilczek construction is that not a self-deformation of 3D-shapes should be represented by motions of a spatial reference frame but the unlocated quantum states should be represented by the motions of “field-shell” in dynamical space-time. Two simple observations may serve as the basis of the intrinsic unifi- cation of relativity and quantum principles. The first observation concerns interference of quantum amplitudes in a fixed quantum setup. A. The linear interference of quantum amplitudes shows the symmetries relative space-time transformations of whole setup. This interference has been studied in “standard” quantum theory. Such symmetries reflects, say, the first order of relativity: the physics is same if any complete setup subject (kinemat- ical, not dynamical!) shifts, rotations, boosts as whole in single Minkowski space-time. According to our notes given above one should add to this list a freely falling quantum setup (super-relativity). The second observation concerns a dynamical “deformation” of some quantum setup. B. If one dynamically changes the setup configuration or its “environ- ment”, then the amplitude of an event will be generally changed. Never- theless there is a different type of tacitly assumed symmetry that may be formulated on the intuitive level as the invariance of physical properties of “quantum particles”, i.e. the invariance of their quantum numbers like mass, spin, charge, etc., relative variation of quantum amplitudes. This means that properties of, say,", + "theless there is a different type of tacitly assumed symmetry that may be formulated on the intuitive level as the invariance of physical properties of “quantum particles”, i.e. the invariance of their quantum numbers like mass, spin, charge, etc., relative variation of quantum amplitudes. This means that properties of, say, physical electrons in two different setups S1 and S2 are the same. One may postulate that the invariant content of this physical properties may be kept if one makes the infinitesimal variation of some “flexible quan- tum setup” reached by a small variation of some fields by adjustment of tuning devices. 6 There is an essential technical approach capable to connect super-relativity and the quantum inertia law. Namely, a new concept of local dynamical variable (LDV) (Leifer, 2004) should be introduced for the realization of infinitesimal variation of a “flexible quantum setup”. This construction is naturally connected with methods developed in studying geometric phase (Berry, 1989). I seek, however, conservation laws for LDV’s in the quantum state space. 3 Invariant classification of quantum motions The idealized separation of a quantum system from the rest of Universe and its cyclic evolution under parameters variation of a Hamiltonian in adia- batic and beyond the adiabatic approximation last time mostly discussed in the framework of geometric phase (Berry, 1989). (Aharonov & Anandan 1987,1988) found non-adiabatic generalization of Berry construction inter- esting in following important direction. Aharonov & Anandan used the gen- eral setting for the geometric phase in terms of connection in fibre bundles over projective Hilbert space. Such approach is important since instead of external parameters of the Hamiltonian, the local projective coordinates of a quantum state itself have been used. For us will be interesting just the invariant classification of quantum motions (Leifer, 1997). These invariant classifications are the quantum analog of classical conditions of the inertial and accelerated motions. They are rooted into the global geometry of the dy- namical group SU(N) manifold. Namely, the geometry of G = SU(N), the isotropy group H = U(1)×U(N −1) of the pure quantum state, and the coset G/H = SU(N)/S[U(1) × U(N −1)] = CP(N −1) geometry, play an essen- tial role in classification of quantum state motions (Leifer, 1997). I assumed that dynamical group of pure internal quantum states |Ψ > of “elemen- tary” quantum particle like electron is G = SU(N) then the isotropy group of |Ψ > in co-moving reference frame is H|Ψ> = {g ∈G : g|Ψ >= |Ψ >} = S[U(1)×U(N−1)] and the coset space G/H|Ψ> = SU(N)/S[U(1)×U(N−1)] will be diffeomorphic to the ray space CP(N −1) that may be treated as state space of internal quantum degrees of freedom. Their self-interacting “eigen-dynamics” will be studied. What is the physical reason of application SU(N) group geometry to the state of motion of quantum particles? Just because we try study the motion of internal degrees of freedom in the space of “unlocated quantum states” CP(N −1). This is the simplest and, probably, the fundamental case is the case of “isolated”, “free” but self-interacting quantum particles in pure quantum state whose", + "of motion of quantum particles? Just because we try study the motion of internal degrees of freedom in the space of “unlocated quantum states” CP(N −1). This is the simplest and, probably, the fundamental case is the case of “isolated”, “free” but self-interacting quantum particles in pure quantum state whose inertial motions could be geometrically analyzed without any reference to the external interaction with a “second particle”. Geometric analysis of quantum state dynamics requires the introduction of a new quantum LDV’s and a new method of their “dynamical identification” by a comparison. I introduce the LDV’s corresponding to the internal SU(N) dynamical group symmetry of quantum states and its breakdown. These LDV’s may be 7 naturally expressed in terms of the following local coordinates πi (j) πi (j) = ( ψi ψj , if 1 ≤i < j ψi+1 ψj if j ≤i < N (1) since SU(N) acts effectively only on the space of rays, i.e. on the classes of equivalence of quantum states differentiating by a non-zero complex multi- plier. LDV’s will be represented by linear combinations of SU(N) generators in local coordinates of CP(N −1) equipped with the Fubini-Study metric Gik∗= [(1 + X |πs|2)δik −πi∗πk](1 + X |πs|2)−2. (2) Hence the internal dynamical variables and their norms should be state- dependent, i.e. local in the state space. These local dynamical variables realize a non-linear representation of the unitary global SU(N) group in the Hilbert state space CN. Namely, N 2 −1 generators of G = SU(N) may be divided in accordance with the Cartan decomposition: [B, B] ∈H, [B, H] ∈B, [B, B] ∈ H. The (N −1)2 generators Φi h ∂ ∂πi + c.c. ∈H, 1 ≤h ≤(N −1)2 (3) of the isotropy group H = U(1)×U(N −1) of the ray and 2(N −1) generators Φi b ∂ ∂πi + c.c. ∈B, 1 ≤b ≤2(N −1) (4) are the coset G/H = SU(N)/S[U(1) × U(N −1)] generators realizing the breakdown of the G = SU(N) symmetry of the generalized coherent states (GCS’s). Here Φi σ, 1 ≤σ ≤N 2 −1 are the coefficient functions of the generators of the non-linear SU(N) realization. They give the infinitesimal shift of the i-component of the coherent state driven by the σ-component of the unitary field exp(iǫλσ) rotating by the generators of AlgSU(N) and they are defined as follows: Φi σ = lim ǫ→0 ǫ−1 \u001a[exp(iǫλσ)]i mψm [exp(iǫλσ)]j mψm −ψi ψj \u001b = lim ǫ→0 ǫ−1{πi(ǫλσ) −πi}, (5) (Leifer, 1997, 2009). Then each of the N 2 −1 generators may be represented by vector fields comprised by the coefficient functions Φi σ contracted with corresponding partial derivatives ∂ ∂πi = 1 2( ∂ ∂ℜπi −i ∂ ∂ℑπi ) and ∂ ∂π∗i = 1 2( ∂ ∂ℜπi + i ∂ ∂ℑπi ). 4 Affine state-dependent gauge fields The anholonomy of the wave function arose due to slowly variable environ- ment was widely discussed by Berry and many other authors in the framework of so-called geometric phases (Berry, 1989). It is clear that now we deal with different problem: Berry made accent on", + "4 Affine state-dependent gauge fields The anholonomy of the wave function arose due to slowly variable environ- ment was widely discussed by Berry and many other authors in the framework of so-called geometric phases (Berry, 1989). It is clear that now we deal with different problem: Berry made accent on variation of wave function during 8 finite cyclic evolution whereas for us interesting the quantum invariants of infinitesimal variation of the quantum setup. The geometric phase is an intrinsic property of the family of eigenstates. There are in fact a set of local dynamical variables (LDV) that like the geo- metric phase intrinsically depends on eigenstates. For us will be interesting vector fields ξk(π1 (j), ..., πN−1 (j) ) : CP(N −1) →C associated with the reaction of quantum state πi (j) on the action of internal “unitary field” exp(iǫλσ) given by Φi σ. In view of the future discussion of infinitesimal unitary transformations, it is useful to compare velocity of variation of the Berry’s phase ˙γn(t) = −An(R) ˙R, (6) where An(R) = ℑ< n(R)|∇Rn(R) > with the affine parallel transport of the vector field ξk(π1, ..., πN−1) given by the equations dξi dτ = −Γ i klξk dπl dτ . (7) The parallel transport of Berry is similar but it is not identical to the affine parallel transport. The last one is the fundamental because this agrees with Fubini-Study “quantum metric tensor” Gik∗in the base manifold CP(N −1). The affine gauge field given by the connection Γ i mn = 1 2Gip∗(∂Gmp∗ ∂πn + ∂Gp∗n ∂πm ) = −δi mπn∗+ δi nπm∗ 1 + P |πs|2 . (8) is of course more close to the Wilczek-Zee non-Abelian gauge fields (Wilczek & Zee, 1984) where the Higgs potential has been replaced by the affine gauge potential (3) whose shape is depicted in Fig. 1. It is involved in the affine 0 1 2 3 4 5 x 0 1 2 3 4 5 y –0.4 –0.3 –0.2 –0.1 0 Fig. 1 The shape of the gauge potential associated with the affine connection in CP(1): Γ = −2 |π| 1+|π|2 , π = x + iy. parallel transport of LDV’s (Leifer, 2005,2007) which agrees with the Fubini- Study metric (2). 9 The transformation law of the connection form Γ i k = Γ i kldπl in CP(N −1) under the differentiable transformations of local coordinates Λi m = ∂πi ∂π′m is as follows: Γ ′i k = Λi mΓ m j Λ−1j k + dΛi sΛ−1s k . (9) It is similar but not identical to well known transformations of non-Abelian fields. The affine Cartan’s moving reference frame takes here the place of “flexible quantum setup”, whose motion refers to itself with infinitesimally close coordinates. Thus we will be rid us of necessity in “second particle” (Anandan & Aharonov, 1988), (Leifer, 1997,2007,2009) as an external refer- ence frame. Such construction perfectly fits for the quantum formulation of the quantum inertia principle (Leifer, 2010) since the affine parallel transport of energy-momentum vector field in CP(N−1) expresses the self-conservation and conditions of stability of,", + "necessity in “second particle” (Anandan & Aharonov, 1988), (Leifer, 1997,2007,2009) as an external refer- ence frame. Such construction perfectly fits for the quantum formulation of the quantum inertia principle (Leifer, 2010) since the affine parallel transport of energy-momentum vector field in CP(N−1) expresses the self-conservation and conditions of stability of, say, electron. 5 State space and dynamical space-time How to lift up the dynamics from the space of “unlocated quantum states” CP(N −1) into the DST or quantum state space cum location? The “in- verse representation” of unitary group SU(N) whose action in the space of internal degrees of freedom CP(N −1) should be realized by a “field shell” motions in dynamical space-time (DST) has been proposed (Leifer, 2010,2009,1997,2007,2004). Thereby, the space-time degrees of freedom and the space-time geometry itself should be derived in order to describe the lump of energy-momentum distribution in the “field shell” wrapping internal degrees of freedom. Let assume that one has, say, a quantum “free” electron with charge and spin and its quantum state is given by the local projective coordinates (π1, π2, π3). Dynamical structure of quantum electron should be accepted seriously since now the inertia principle refers just to internal quantum state without evident reference to space-time coordinates. Hence one need initially to deal with the dynamics of quantum degrees of freedom in CP(3). Notice, I am not intended here to establish the connection of this model with the Standard Model. This will be a next step of the investigation. The distance between two quantum states of electron in CP(3) given by the Fubibi-Study invariant interval dSF.−S. = Gik∗dπidπk∗. The speed of the interval variation is given by the equation (dSF.−S. dτ )2 = Gik∗ dπi dτ dπk∗ dτ = c2 ℏ2 Gik∗(Φi µP µ)(Φk∗ ν P ν∗) (10) relative “quantum proper time” τ where energy-momentum vector field P µ(x) obeys field equations that will be derived later. This internal dynamics should be expressed in space-time coordinates xµ assuming that variation of coordinates δxµ arise due to the transformations of Lorentz reference frame “centered” about covariant derivative δP ν δτ = δxµ δτ ( ∂P ν ∂xµ + Γ ν µλP λ) in dynam- ical space-time (DST). Such procedure may be called “inverse representa- tion” (Leifer, 2010,2009) since this intended to represent quantum motions 10 in CP(3) by “quantum Lorentz transformation” in DST as will be described below. Local Lorentz frame will be built on the basis of the qubit spinor η whose components may be locally (in CP(3)) adjusted by “quantum boosts” and “quantum rotations” so that the velocity of the spinor variation coincides with velocity variation of the Jacobi vector fields: tangent Jacobi vector field η0 = Jtang(π) = (aiτ + bi)U i(π) giving initial frequencies traversing the geodesic and the initial phases, and the normal Jacobi vector field η1 = Jnorm(π) = [ci sin(√κτ) + di cos(√κτ)]U i(π) showing deviation from one geodesic to the another one (Besse, 1978). Thereby, two invariantly separated motions have been taken into account: 1)“free motion” of spin/charge degrees of freedom along energy-momentum tangent vector to geodesic", + "and the normal Jacobi vector field η1 = Jnorm(π) = [ci sin(√κτ) + di cos(√κτ)]U i(π) showing deviation from one geodesic to the another one (Besse, 1978). Thereby, two invariantly separated motions have been taken into account: 1)“free motion” of spin/charge degrees of freedom along energy-momentum tangent vector to geodesic line in CP(3) (oscillation of massive mode in the vicinity of a minimum of the affine potential across its valley) generated by the coset transformations G/H = SU(4)/S[U(1) × U(3)] = CP(3) and, 2) deviation of geodesic line motion in the direction of normal Jacobi vector field transversal to reference geodesic line (oscillation of massless mode along the valley of the affine potential) generated by the isotropy group H = U(1) × U(3). The sufficiency condition of the functional extremum has been applied here to the action for the self-interacting electron. Namely, variation of geodesics in CP(3) represented by the Jacobi fields may be written as the Hamilton equations (Sternberg, 1964) for energy-momentum taking the place of the gauge potentials in DST. This field should serve as a generalization of elec- tromagnetic field since Jacobi equations for variation of geodesics in the sym- metric space CP(N −1) with constant sectional curvature κ is in fact the equations for the coupled harmonic oscillators. The sufficiency conditions use the concepts of second Lagrangian, second Hamiltonian and second extremal (Young, 1969) without, however, evident relations to the method of second quantization. The sectional curvature κ of CP(3) is not specified yet. This “measure- ment” means that in the DST only deviations from the geodesic motion is “observable” due to electromagnetic-like field surrounding electron. This field may be mapped onto DST if one assumes that transition from one GCS of the electron to another is accompanied by dynamical transition from one Lorentz frame to another. Thereby, infinitesimal Lorentz transformations define small DST due to coordinate variations δxµ. It is convenient to take Lorentz transformations in the following form ct′ = ct + (xaQ)δτ x′ = x + ctaQδτ + (ωQ × x)δτ (11) where I put for the parameters of quantum acceleration and rotation the definitions aQ = (a1/c, a2/c, a3/c), ωQ = (ω1, ω2, ω3) (Misner, Thorne, Wheeler, 1973) in order to have for τ the physical dimension of time. The expression for the “4-velocity” V µ is as follows V µ Q = δxµ δτ = (xaQ, ctaQ + ωQ × x). (12) 11 The coordinates xµ of imaging point in dynamical space-time serve here merely for the parametrization of the energy-momentum distribution in the “field shell” described by quasi-linear field equations (Leifer, 2009,2010) that will be derived below. Any two infinitesimally close spinors η and η + δη may be formally con- nected with infinitesimal SL(2, C) transformations represented by “Lorentz spin transformations matrix” (Misner, Thorne, Wheeler, 1973) ˆL = \u0012 1 −i 2δτ(ω3 + ia3) −i 2δτ(ω1 + ia1 −i(ω2 + ia2)) −i 2δτ(ω1 + ia1 + i(ω2 + ia2)) 1 −i 2δτ(−ω3 −ia3) \u0013 .(13) Then “quantum accelerations” a1, a2, a3 and “quantum angular velocities” ω1, ω2, ω3 may be found", + "matrix” (Misner, Thorne, Wheeler, 1973) ˆL = \u0012 1 −i 2δτ(ω3 + ia3) −i 2δτ(ω1 + ia1 −i(ω2 + ia2)) −i 2δτ(ω1 + ia1 + i(ω2 + ia2)) 1 −i 2δτ(−ω3 −ia3) \u0013 .(13) Then “quantum accelerations” a1, a2, a3 and “quantum angular velocities” ω1, ω2, ω3 may be found in the linear approximation from the equation δη = ˆLη −η and from the equations for the velocities ξ of η spinor variations expressed in two different forms: ˆR \u0012 η0 η1 \u0013 = 1 δτ (ˆL −ˆ1) \u0012 η0 η1 \u0013 = \u0012 ξ0 ξ1 \u0013 (14) and \u0012 ξ0 ξ1 \u0013 = δ{(aiτ+bi)Ui(π)} δτ δ{[ci sin(√κτ)+di cos(√κτ)]Ui(π)} δτ ! (15) Since CP(3) is totally geodesic manifold (Kobayashi & Nomizu, 1969), each geodesic belongs to some CP(1) parameterized by the single complex variable π = e−iφ tan(θ/2). Then the tangent vector field U(π) = δπ δτ = ∂π ∂θ δθ δτ + ∂π ∂φ δφ δτ , where δθ δτ = −ω3 sin(θ) −((a2 + ω1) cos(φ) + (a1 −ω2) sin(φ)) sin(θ/2)2 −((a2 −ω1) cos(φ) + (a1 + ω2) sin(φ)) cos(θ/2)2; δφ δτ = a3 + (1/2)(((a1 −ω2) cos(φ) −(a2 + ω1) sin(φ)) tan(θ/2) −((a1 + ω2) cos(φ) −(a2 −ω1) sin(φ)) cot(θ/2)), (16) will be parallel transported, i.e. U ′(π) = δ2π δτ 2 = 0. The linear system of 6 real non-homogeneous equation ℜ( ˆR00η0 + ˆR01η1) = ℜ(ξ0), ℑ( ˆR00η0 + ˆR01η1) = ℑ(ξ0), ℜ( ˆR10η0 + ˆR11η1) = ℜ(ξ1), ℑ( ˆR10η0 + ˆR11η1) = ℑ(ξ1), δθ δτ = F1, δφ δτ = F2, (17) gives the “quantum boost” aQ(ℜ(η0), ℑ(η0), ℜ(η1), ℑ(η1), θ, φ), and “quan- tum rotation” ωQ(ℜ(η0), ℑ(η0), ℜ(η1), ℑ(η1), θ, φ). 12 The components of the spinor (ℜ(η0), ℑ(η0), ℜ(η1), ℑ(η1)) should be agreed with the coefficients (a, b, c, d) into (24) by the condition of solvability of the system (24) reads as Det|...| = 0 for the extended matrix |...|. Two frequencies (F1, F2) will be found from the spectrum of excitations discussed below. One frequency gives the coset deformation acting along some geodesic in CP(3) and the second one gives the velocity of rotation of the geodesic under the action of the gauge isotropy group H = U(1) × U(3). 6 Self-interacting quantum electron I assume that the spin/charge quantum state of free electron and similar quantum states of freely falling electron should be physically identical. Dy- namical equivalence of these quantum states will expressed by the conser- vation of energy-momentum vector field of quantum electron most naturally may be realized by their affine parallel transport in CP(3). Therefore, if the principle of weak equivalence is valid, then the plane wave cannot be the true quantum state of the free electron and one should derive a new non-linear equation for the description of its “field-shell”. In fact, the requirement of the affine parallel transport puts strong restriction on the “field-shell” sup- porting cyclic spin/charge degrees of freedom motion along close geodesic of CP(3). The local projective coordinates coordinates of eigenstate πi (j) = ( ψi ψj , if 1 ≤i < j", + "of its “field-shell”. In fact, the requirement of the affine parallel transport puts strong restriction on the “field-shell” sup- porting cyclic spin/charge degrees of freedom motion along close geodesic of CP(3). The local projective coordinates coordinates of eigenstate πi (j) = ( ψi ψj , if 1 ≤i < j ψi+1 ψj if j ≤i < 4 (18) in the map Uj : {|Ψ >, |ψj| ̸= 0}, 1 ≤j ≤4 of free electron in CP(3) may be derived from ordinary homogeneous system of eigen-problem. It is easy to see (Leifer, 2007) that under transition from the system of homogeneous equations to the reduced system of non-homogeneous equations (the first equation was omitted) (−E + mc2)π1 + c(px + ipy)π2 −cpzπ3 = 0 c(px −ipy)π1 −(E + mc2)π2 = cpz −cpzπ1 −(E + mc2)π3 = c(px + ipy), (19) one has the single-value solution for eigen-ray π1 = 0, π2 = −cpz E + mc2 π3 = −c(px + ipy) E + mc2 , (20) in the map U1 : {ψ1 ̸= 0} for E = p m2c4 + c2p2 ± ∆. It is possible only if the determinant of the reduced system D = (E2 −m2c4 −c2p2)2 is not equal zero. It is naturally to use these scale-invariant functional variables (π1, π2, π3) in order to establish relation between spin-charge degrees of free- dom and energy-momentum distribution of electron in dynamical space-time (DST) since the “off-shell” condition D = (E2 −m2c4 −c2p2)2 ̸= 0 opens the way for self-interaction. New dispersion law will be established due to 13 formulation of the conservation law of quantum energy-momentum. In local coordinates (representation) the improper states like plane waves are simply deleted. It means that trivial free motion of whole quantum setup in local homogeneous space-time is removed. One may treat this approach as “self- interacting representation” where only self-interacting effects should be taken into account. Then the dynamics in state space CP(3) plays the leading role whereas a field dynamics in space-time will be derivable. Infinitesimal energy-momentum variations evoked by interaction charge- spin degrees of freedom (implicit in ˆγµ ) that may be expressed in terms of local coordinates πi since there is a diffeomorphism between the space of the rays CP(3) and the SU(4) group sub-manifold of the coset transformations G/H = SU(4)/S[U(1) × U(3)] = CP(3) and the isotropy group H = U(1) × U(3) of some state vector. It will be expressed by the combinations of the SU(4) generators ˆγµ of unitary transformations that will be defined by an equation arising under infinitesimal variation of the energy-momentum Φi µ(γµ) = lim ǫ→0 ǫ−1 \u001a[exp(iǫˆγµ)]i mψm [exp(iǫˆγµ)]j mψm −ψi ψj \u001b = lim ǫ→0 ǫ−1{πi(ǫˆγµ) −πi}, (21) arose in a nonlinear local realization of SU(4) (Leifer, 2009). Here ψm, 1 ≤ m ≤4 are the ordinary bi-spinor amplitudes. I calculated the twelve coeffi- cient functions Φi µ(γµ) in the map U1 : {ψ1 ̸= 0}: Φ1 0(γ0) = 0, Φ2 0(γ0) = −2iπ2, Φ3 0(γ0) = −2iπ3; Φ1 1(γ1) = π2 −π1π3, Φ2 1(γ1) = −π1 −π2π3, Φ3 1(γ1) = −1", + "m ≤4 are the ordinary bi-spinor amplitudes. I calculated the twelve coeffi- cient functions Φi µ(γµ) in the map U1 : {ψ1 ̸= 0}: Φ1 0(γ0) = 0, Φ2 0(γ0) = −2iπ2, Φ3 0(γ0) = −2iπ3; Φ1 1(γ1) = π2 −π1π3, Φ2 1(γ1) = −π1 −π2π3, Φ3 1(γ1) = −1 −(π3)2; Φ1 2(γ2) = i(π2 + π1π3), Φ2 2(γ2) = i(π1 + π2π3), Φ3 2(γ2) = i(−1 + (π3)2); Φ1 3(γ3) = −π3 −π1π2, Φ2 3(γ3) = −1 −(π2)2, Φ3 3(γ3) = π1 −π2π3. (22) Now I will define the Γ-vector field Γµ = Φi µ(π1, π2, π3) ∂ ∂πi (23) and then the energy-momentum operator will be defined as the functional vector field P µΓµΨ(π1, π2, π3) = P µΦi µ(π1, π2, π3) ∂ ∂πi Ψ(π1, π2, π3) + c.c. (24) acting on the “total wave function”, where the ordinary 4-momentum P µ = ( E c −e cφ, P−e cA) = ( ℏω c −e cφ, ℏk−e cA) (not operator-valued) should be identi- fied with the solution of quasi-linear “field-shell” PDE’s for the contravariant components of the energy-momentum tangent vector field in CP(3) P i(x, π) = P µ(x)Φi µ(π1, π2, π3), (25) where P µ(x) is energy-momentum distribution that comprise of “field-shell” of the self-interacting electron. One sees that infinitesimal variation of energy-momentum is represented by the operator of partial differentiation in complex local coordinates πi with corresponding coefficient functions Φi µ(π1, π2, π3). Then the single-component 14 “total wave function” Ψ(π1, π2, π3) should be studied in the framework of new PDE instead of two-component approximation due to Foldy-Wouthuysen uni- tary transformations. There are of course four such functions Ψ(π1 (1), π2 (1), π3 (1)), Ψ(π1 (2), π2 (2), π3 (2)), Ψ(π1 (3), π2 (3), π3 (3)), Ψ(π1 (4), π2 (4), π3 (4)) - one function in each local map. The “field-shell” equations may be derived as the consequence of the conservation law of the energy-momentum (Leifer, 2009). In the reply on questions of some colleagues (why, say, Lagrangian is not used for derivation of the field equations?) I would like to note following. Strictly speaking the least action principle is realized only in average that is clear from Feynman’s summation of quantum amplitudes. Hence one may suspect that more deep principle should be used for derivation of fundamental equations of motion. The quantum formulation of the inertia law has been used (Leifer, 2010). What the inertial principle means for quantum systems and their states? Formally the inertial principle is tacitly accepted in the package with rel- ativistic invariance. But we already saw that the problem of identification and therefore the localization of quantum particles in classical space-time is problematic and it requires a clarification. I assumed that quantum version of the inertia law may be formulated as follows: inertial quantum motion of quantum system may be expressed as a self-conservation of its local dynamical variables like energy- momentum, spin, charge, etc. The conservation law of the energy-momentum vector field in CP(3) dur- ing inertial evolution will be expressed by the equation of the affine parallel transport δ[P", + "follows: inertial quantum motion of quantum system may be expressed as a self-conservation of its local dynamical variables like energy- momentum, spin, charge, etc. The conservation law of the energy-momentum vector field in CP(3) dur- ing inertial evolution will be expressed by the equation of the affine parallel transport δ[P µΦi µ(γµ)] δτ = 0, (26) which is equivalent to the following system of four coupled quasi-linear PDE for dynamical space-time distribution of energy-momentum “field-shell” of quantum state V µ Q(∂P ν ∂xµ + Γ ν µλP λ) = −c ℏ(Γ m mnΦn µ(γ) + ∂Φn µ(γ) ∂πn )P νP µ, (27) and ordinary differential equations for relative amplitudes dπk dτ = c ℏΦk µP µ, (28) which is in fact the equations of characteristic for linear “super-Dirac” equa- tion iP µΦi µ(γµ) ∂Ψ ∂πi = mcΨ (29) that supposes ODE for single “total state function” iℏdΨ dτ = mc2Ψ (30) with the solution for variable mass m(τ) Ψ(T ) = Ψ(0)e−i c2 ℏ R T 0 m(τ)dτ. (31) 15 Probably, simple relation given by Fubini-Study metric for the square of the frequency Ω2 = (dSF.−S. dτ )2 = Gik∗ dπi dτ dπk∗ dτ = c2 ℏ2 Gik∗(Φi µP µ)(Φk∗ ν P ν) (32) associated with velocity traversing geodesic line (6.14) during spin/charge variations in CP(3) sheds the light on the mass problem of self-interacting electron in the connection with action principle on CP(3). Taking into ac- count the “off-shell” dispersion law ℏ2 c2 Gik∗ dπi dτ dπk∗ dτ = Gik∗(Φi µP µ)(Φk∗ ν P ν∗) = (Gik∗Φi µΦk∗ ν )P µP ν∗= gµνP µP ν∗= m2c2 (33) one has idΨ dτ = mc2 ℏΨ = Ψ r Gik∗ dπi dτ dπk∗ dτ = Ψ q dS2 F.−S. = ±ΨdSF.−S. (34) and, therefore, Ψ(T ) = Ψ(0)e±iSF.−S.. (35) In other words dSF.−S. = c2 ℏmdτ. (36) The metric tensor of the local DST in the vicinity of electron gµν = Gik∗Φi µΦk∗ ν is state dependable and it will be discussed in a separated article. The system of quasi-liner PDE’s (6.10) following from the conservation law, ODE’s and algebraic linear non-homogeneous equations comprise of the self-consistent problem for stability (in fact - existing) of electron. Their solu- tion have been shortly discussed (Leifer, 2009). The theory of these equations is well known (Courant & Hilbert, 1989). One has the quasi-linear PDE sys- tem with identical principle part V µ Q for which we will build system of ODE’s of characteristics dxν dτ = V ν Q, dP ν dτ = −V µ QΓ ν µλP λ −c ℏ(Γ m mnΦn µ(γ) + ∂Φn µ(γ) ∂πn )P νP µ, dπk dτ = c ℏΦk µP µ. (37) One of the integrable combination is as follows δx0 V 0 Q = δP 0 P 0(L0P 0 + L1P 1 + L2P 2 + L3P 3), (38) where Lµ = −c ℏ(Γ m mnΦn µ(γ) + ∂Φn µ(γ) ∂πn ). If L0P 0 < 0 then one has implicit solution x0 aαxα + t0 = − 2 LαP α tanh−1(1 + 2L0P 0 LαP α", + "0(L0P 0 + L1P 1 + L2P 2 + L3P 3), (38) where Lµ = −c ℏ(Γ m mnΦn µ(γ) + ∂Φn µ(γ) ∂πn ). If L0P 0 < 0 then one has implicit solution x0 aαxα + t0 = − 2 LαP α tanh−1(1 + 2L0P 0 LαP α ), (39) 16 where t0 is an integration constant. An explicit solution for energy is the kink P 0 = LαP α 2L0 [tanh(−( x0 aαxα + t0)LαP α 2 ) −1]. (40) If I put LαP α 2 = 1, L0 = 1, V = aαxα = 0.6 the kink solution may be represented by the graphic in Fig. 2. This solution represent the lump of self- Fig. 2 The kink solution of the quasi-linear PDE’s in dynamical space-time show- ing the distribution of energy-momentum “field-shell” of extended quantum elec- tron. It is not solution of a runaway type. interacting electron through an electro-magnetic-like field in the co-moving Lorentz reference frame. The nature of this field will be discussed in a sep- arate article. In the standard QED self-interacting effects are treated as a polarization of the vacuum. In the present picture the lump is dynamically self-supporting by outward and inward waves whose characteristics are rep- resented by the equations (6.21). 7 Stability of energy-momentum characteristics and dispersion law Let me discuss the system of the characteristics dP ν dτ = −V µ QΓ ν µλP λ −c ℏ(Γ m mnΦn µ(γ) + ∂Φn µ(γ) ∂πn )P νP µ. (41) Their stability will provide the equilibrium of the distribution of energy- momentum in DST. The standard approach to stability analysis instructs us to find the stationary points. The stationary condition δP λ δτ = 0 (42) leads to the system of algebraic equations V µ QΓ ν µλP λ + c ℏ(Γ m mnΦn µ(γ) + ∂Φn µ(γ) ∂πn )P νP µ = 0. (43) 17 The probing solution in the vicinity of the stationary points P µ 0 is as follows P µ(τ) = P µ 0 + pµeωτ. (44) This solution being substituted in the equation (7.3) gives the homogeneous linear system ℏω c pν + ℏ c V µ QΓ ν µλpλ + (Γ m mnΦn µ(γ) + ∂Φn µ(γ) ∂πn )pµP ν 0 = 0. (45) The determinant of this system is as follows D1 = (ℏω c )4 + α(ℏω c )3 + β(ℏω c )2 + γ(ℏω c ) + δ, (46) with complicated coefficients α, β, γ, δ. I put Kν λ = ℏ c V µ QΓ ν µλ and M ν µ = (Γ m mnΦn µ(γ) + ∂Φn µ(γ) ∂πn )P ν 0 then one may find that α = T r(Kν λ) + T r(M ν µ) (47) and β = K0 0(L1P 1 0 + L2P 2 0 + L3P 3 0 ) + K1 1(L0P 0 0 + L2P 2 0 + L3P 3 0 ) +K2 2(L1P 1 0 + L0P 0 0 + L3P 3 0 ) + K3 3(L1P 1 0 + L0P 0 0", + "= K0 0(L1P 1 0 + L2P 2 0 + L3P 3 0 ) + K1 1(L0P 0 0 + L2P 2 0 + L3P 3 0 ) +K2 2(L1P 1 0 + L0P 0 0 + L3P 3 0 ) + K3 3(L1P 1 0 + L0P 0 0 + L2P 2 0 ) −K0 1L0P 1 0 −K1 0L1P 0 0 −K0 2L0P 2 0 −K2 0L2P 0 0 −K0 3L0P 3 0 −K3 0L3P 0 0 −K1 2L1P 2 0 −K2 1L2P 1 0 −K1 3L1P 3 0 −K3 1L3P 1 0 −K2 3L2P 3 0 −K3 2L3P 2 0 , (48) whereas γ, δ have higher order in ℏ c and they may be temporarily discarded in approximate dispersion law. This dispersion law may be written as follows (ℏω c )2[(ℏω c )2 + α(ℏω c ) + β] = 0. (49) The trivial solution ω1,2 = 0 has already been discussed (Leifer, 2009). Two non-trivial solutions when α2 ≫β are given by the equations ℏω3,4 = cα −1 ± q 1 −4β α2 2 ≈cα−1 ± (1 −2β α2 ) 2 ; ℏω3 = −cβ α , ℏω4 = −cα + cβ α . (50) Both parameters α, β are in fact complex functions of (π1, π2, π3) but they are linear in stationary momenta P µ 0 . Coefficients Kν µ ∝ℏ c are much smaller than M ν µ. The negative real part of these two roots substituted in the probing function (7.4) will define attractors and two finite masses. One should find solution of the non-linear system (7.3). Its approximate solution in the vicinity of P µ test = (mc2, 0, 0, 0) has been found by the method of Newton: P µ 0 = P µ test + δµ + ..., (51) 18 where δµ is the solution of the Newton’s first approximation equations (2L0mc + K0 0)δ0 + (L1mc + K0 1)δ1+ (L2mc + K0 2)δ2 + (L3mc + K0 3)δ3 = −(L0m2c4 + K0 0mc3) c2 K1 0δ0 + (L0mc + K1 1)δ1 + K1 2δ2 + K1 3δ3 = −K1 0mc K2 0δ0 + K2 1δ1 + (L0mc + K2 2)δ2 + K2 3δ3 = −K2 0mc K3 0δ0 + K3 1δ1 + K3 2δ2 + (L0mc + K3 3)δ3 = −K3 0mc, (52) where Lµ = (Γ m mnΦn µ(γ) + ∂Φn µ(γ) ∂πn ) is now dimensionless. The solution of this system is as follows: δ0 = −mc2 mcL2 0 −K1 0L1 −K2 0L2 −K3 0L3 2mcL2 0 −K1 0L1 −K2 0L2 −K3 0L3 δ1 = −mc K1 0L1 2mcL2 0 −K1 0L1 −K2 0L2 −K3 0L3 δ2 = −mc K2 0L2 2mcL2 0 −K1 0L1 −K2 0L2 −K3 0L3 δ3 = −mc K3 0L3 2mcL2 0 −K1 0L1 −K2 0L2 −K3 0L3 , (53) where I took into account Kµ ν + Kν µ = 0. If hypothesis about dynamical nature of electron mass defined by self- interacting spin/charge degrees of freedom is correct then it is very natural to assume that F1 = δθ δτ =", + "0L1 −K2 0L2 −K3 0L3 , (53) where I took into account Kµ ν + Kν µ = 0. If hypothesis about dynamical nature of electron mass defined by self- interacting spin/charge degrees of freedom is correct then it is very natural to assume that F1 = δθ δτ = ℜ(ω3) = c ℏℜ(−β α ), or F1 = δθ δτ = ℜ(ω4) = c ℏℜ(−α + β α), and F2 = δφ δτ = ℑ(ω3) = c ℏℑ(−β α ), or F2 = δφ δτ = ℑ(ω4) = c ℏℑ(−α + β α). (54) The conditions of instability are given by the simple inequalities: ℜ(α)ℜ(β)+ ℑ(α)ℑ(β) < 0 for ω3 and ℜ(α)ℜ(β) + ℑ(α)ℑ(β) −ℜ(α)|α|2 > 0 for ω4. However, the solution of complicated self-consistent problem (5.8), (6.21), (7.9) and (7.13), (7.14) is not found up to now. Therefore the problem of dynamically generated mass is not solved yet. But the “field-shell” of the self-interacting quantum electron may be interesting in the connection with old problem of the runaway solution (see (Hammond, 2010) and references therein). Contradictable field structure of quantum electron requires some clarifications. Let me start with simplified equation (7.1) in the vicinity of the point (π1 = π2 = π3 = 0). The equation of characteristics will be treated here as the equations of motion for the self-momentum dP ν dτ = −c ℏKν µP µ + 4ic ℏP 0P ν, (55) 19 where the linear term contains the matrix Kν µ = ℏ c2 ax    0 −a1 −a2 −a3 a1 0 −ω3 ω2 a2 ω3 0 −ω1 a3 −ω2 ω1 0    (56) and the non-linear term is imaginary since Lµ(0) = (−4i, 0, 0, 0). The right part linearized in P µ is the Jacobi matrix Jν µ =     8ic ℏP 0 −K0 1 −K0 2 −K0 3 −K1 0 + 4ic ℏP 1 4ic ℏP 0 −K1 2 −K1 3 −K2 0 + 4ic ℏP 2 −K2 1 4ic ℏP 0 −K2 3 −K3 0 + 4ic ℏP 3 −K3 1 −K3 2 4ic ℏP 0     (57) computed at the stationary point (7.11). If one puts P µ 0 = (0, 0, 0, 0), and for simplicity a2 = a3 = ω2 = ω3 = 0 then one has a neutral stability since eigen values of Jν µ are λ1 = iω1, λ2 = −iω1, λ3 = ia1, λ4 = −ia1. If P µ 0 = P µ test = (mc2, 0, 0, 0) then eigen values will be as follows: λ1 = i(6m + p 4m2 + a2 1), λ2 = i(6m − p 4m2 + a2 1), λ3 = i(4m + ω1), λ4 = i(4m −ω1). If, however, one takes into account the first approximation corrections (7.13) then one get λ1 = i[6(m + δ0) + p 4(m + δ0)2 + a2 1 −4ia1δ1], λ2 = i[6(m + δ0)− p 4(m + δ0)2 + a2 1 −4ia1δ1], λ3 = i(4m+ω1 +δ0), λ4 = i(4m−ω1 +δ0) showing possible bifurcation instability if", + "into account the first approximation corrections (7.13) then one get λ1 = i[6(m + δ0) + p 4(m + δ0)2 + a2 1 −4ia1δ1], λ2 = i[6(m + δ0)− p 4(m + δ0)2 + a2 1 −4ia1δ1], λ3 = i(4m+ω1 +δ0), λ4 = i(4m−ω1 +δ0) showing possible bifurcation instability if the mass m used as a bifurcation parameter. Analysis of the full Riccati system (7.1) and even its linearized version is too complicated and it is not finished. One may, however, find the exact solutions of (7.15) in two separated cases: (a1 ̸= 0, a2 = a3 = ω1 = ω2 = ω3 = 0) and (ω1 ̸= 0, a1 = a2 = a3 = ω2 = ω3 = 0) where I assumed that (a1 ̸= 0, ω1 ̸= 0) are not the field solution of the full self-consistent problem but simply constant number parameters (I’ve put ax = 1). Even oversimplified results given merely for illustration show a rich internal dynamical structure of self-interacting electron. Exact solution of (7.15) will be represented graphically with needed ex- planations. A. Let me assume (a1 = 2, a2 = a3 = ω1 = ω2 = ω3 = 0), i.e. x = 1/2. The boost parameter a1 has the dimension of a momentum here, and four integration constants have following physical dimensions: [C1 = 3] = momentum, [C2 = 1] = momentum, [C3 = 5] = dimensionless, [C4 = 3] = dimensionless. P 0(τ) = ia1[C4 cos( a1cτ ℏ) −sin( a1cτ ℏ)] 4[C3 + C4 sin( a1cτ ℏ) −cos( a1cτ ℏ)] (58) P 3(τ) = C2 C3 + C4 sin( a1cτ ℏ) −cos( a1cτ ℏ) (59) One sees that the boost of the Lorentz frame attached to GCS of the electron leads to periodic oscillations of all components of energy-momentum. 20 Fig. 3 The imaginary part of the energy P 0(τ) under the boost a1 = 2 in x- direction. Fig. 4 The real part of the z-momentum P 3(τ) under the boost a1 = 2 in x- direction. B. Let me assume now (ω1 = 1, a1 = a2 = a3 = ω2 = ω3 = 0). The rotation parameter ω1 has here the dimension of a momentum, and four integration constants have following physical dimensions: [C1 = 3] = momentum, [C2 = 1] = momentum−1, [C3 = 5] = action, [C4 = 3] = action. P 0(τ)re = ℜ( ℏ C2ℏ−4icτ ) (60) Fig. 5 The real part of the energy P 0(τ)re under the rotation ω1 = 1 about axes OX. P 0(τ)im = ℑ( ℏ C2ℏ−4icτ ) (61) P 2(τ)re = ℜ(−C3 sin( ω1cτ ℏ) + C4 cos( ω1cτ ℏ) C2ℏ−4icτ ) (62) 21 Fig. 6 The imaginary part of the energy P 0(τ)im under the rotation ω1 = 1 about axes OX. Fig. 7 The real part of the y-momentum P 2(τ)re under the rotation ω1 = 1 about axes OX. P 2(τ)im = ℑ(−C3 sin( ω1cτ ℏ) + C4 cos( ω1cτ ℏ) C2ℏ−4icτ ) (63) One sees that rotations of the Lorentz frame attached to GCS of the elec-", + "axes OX. Fig. 7 The real part of the y-momentum P 2(τ)re under the rotation ω1 = 1 about axes OX. P 2(τ)im = ℑ(−C3 sin( ω1cτ ℏ) + C4 cos( ω1cτ ℏ) C2ℏ−4icτ ) (63) One sees that rotations of the Lorentz frame attached to GCS of the elec- Fig. 8 The imaginary part of the y-momentum P 2(τ)im under the rotation ω1 = 1 about axes OX. tron describe the radiation force leading to irreversible loss of energy and momentum. Generally, the oversimplified equation (7.15) leads to finite solu- tions for all components of the energy-momentum, i.e. no runaway solutions for self-interacting electron. I suspect that self-consistent solution may lead only to more fast decay of the momentum as a function of deviation from stationary state. In the previous examples the “proper quantum time” τ is in fact the measure of such deviation in CP(3). 22 8 Conclusion Analysis of the localization problem insists to make attempts of the intrin- sic unification of quantum principles based on the fundamental concept of quantum amplitudes and the principle of relativity ensures the physical equiv- alence of any conceivable quantum setup. The realization of such program evokes the necessity of the state-dependent affine gauge field in the state space that acquires reliable physical basis under the quantum formulation of the inertia law (self-conservation of local dynamical variables of quantum particle during inertial motion). Representation of such affine gauge field in dynam- ical space-time has been applied to the relativistic extended self-interacting Dirac’s electron (Leifer, 2009,2010). Thus one has unconstrainedly given lo- calized (soliton-like) solution and promising dispersion law with a mass gap. The simplest formulation of the quantum inertia law by the affine parallel transport of energy-momentum in the projective Hilbert state space has been proposed (Leifer, 2010). Such approach paves the way to clarification of the old problem of inertial mass and such “fictitious” forces as, say, centrifugal force. Shortly speaking the inertia and inertial forces are originated not in space-time but it the space of quantum states since they are generated by the deformation of quantum states as a reaction on an external influence. References 1. Aharonov Y. et al, Mearurement of Time-Arrival in Quantum Mechanics, Phys. Rev., A57,4130 (1998). 2. Anandan J., Aharonov Y., Phys. Rev. D, 38, (6) 1863-1870 (1988). 3. Berry M.V., “The Quantum Phase, Five Years After” in Geometric Phases in Physics, World Scientific. 1989. 4. Besse A.L., Manifolds all of whose Geodesics are Closed, Springer-Verlag, Berlin, Heidelberg, New Yourk, (1978). 5. Courant R., Hilbert D., Methods of Mathematical Physics, 2, Partial Differential Equations, 830, Wiley, (1989) 6. Dirac P.A.M., The principls of quantum mechanics, Fourth Edition, Oxford, At the Clarebdon Press, 1958. 7. Einstein A., Ann. Phys. Zur Electrodynamik der bewegter K¨orper, 17, 891-921 (1905). 8. Einstein A., Letter to Schr¨odinger from 22. XII. 1950. 9. Einstein A., “Einleitende Bemerkungen ¨uber Grundbegriffe” in “Louis de Broglie, physicien et penseur”, Paris, pp. 4-14, (1953). 10. Einstein A., Ann. Phys. Die Grunlage der allgemeinen Relativit¨atstheorie, 49, 769-822 (1916). 11. Foldy L.L., Wouthuysen S.A., Phys. Rev., 78, 29 (1950).", + "Einstein A., Letter to Schr¨odinger from 22. XII. 1950. 9. Einstein A., “Einleitende Bemerkungen ¨uber Grundbegriffe” in “Louis de Broglie, physicien et penseur”, Paris, pp. 4-14, (1953). 10. Einstein A., Ann. Phys. Die Grunlage der allgemeinen Relativit¨atstheorie, 49, 769-822 (1916). 11. Foldy L.L., Wouthuysen S.A., Phys. Rev., 78, 29 (1950). 12. Hammond R.T., Relativistic Particle Motion and Radiation Reaction in Elec- trodynamics, 2010, EJTP 7, No. 23, 221-258. 13. Kobayashi S., and Nomizu K.,a 414, Foundations of Differential Geometry, V. II, Interscience Publishers, New York-London-Sydney, (1969). 14. Leifer P., The quantum content of the inertia law and field dynamics, arXiv:1009.5232v1. 15. Leifer P., Self-interacting quantum electron, arXiv:0904.3695v3. 16. Leifer P., Superrelativity as an Element of a Final Theory, Found. Phys. 27, (2) 261 (1997). 17. Leifer P., Objective quantum theory based on the CP(N −1) affine gauge field, Annales de la Fondation Louis de Broglie, 32, (1) 25-50 (2007). 23 18. Leifer P., State-dependent dynamical variables in quantum theory, JETP Let- ters, 80, (5) 367-370 (2004). 19. Leifer P., Inertia as the “threshold of elasticity” of quantum states, Found.Phys.Lett., 11, (3) 233 (1998). 20. Leifer P., An affine gauge theory of elementary particles, Found.Phys.Lett., 18, (2) 195-204 (2005). 21. Misner C.W., Thorne K.S., Wheeler J.A.,Gravitation, 1279, W.H.Freeman and Company, San Francisco (1973). 22. Newton T.D. and Wigner E.P., Localized States for Elementary Systems, Rev. Mod. Phys., 21, No.3, 400-406 (1949). 23. Shapere A. & Wilczek F., Geometry of self-propolsion at low Reynolds number, J. Fluid Mech. (1989), vol. 198, pp 557-585. 24. Sterberg S., Lectures on differential geometry, Prentice Hall, Inc., Englewood Cliffs, N.J. (1964). 25. Umezawa H., Matsumoto H., Tachiki M., Thermo Field Dynamics and Con- densed States, North-Holland Publishing Company, Amsterdam, New York, Ox- ford, 504 (1982). 26. Wilczek F. & Zee A., Phys. Rev. Lett., 52, (24) 2111-2114 (1984). 27. Young L.C., Lectures on the calculus of variations and optimal control theory, W.B. Saunders Company, Philadelphia, London, Toronto, (1969).", + "physica status solidi Electron quantum optics as quantum signal processing B. Roussel1, C. Cabart1, G. F`eve2, E. Thibierge1, P. Degiovanni1 1 Univ Lyon, Ens de Lyon, Universit´e Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France 2 Laboratoire Pierre Aigrain, Ecole Normale Sup´erieure-PSL Research University, CNRS, Universit´e Pierre et Marie Curie-Sorbonne Universit´es, Universit´e Paris Diderot-Sorbonne Paris Cit´e, 24 rue Lhomond, 75231 Paris Cedex 05, France ∗Corresponding author: mail: Pascal.Degiovanni@ens-lyon.fr The recent developments of electron quantum optics in quantum Hall edge channels have given us new ways to probe the behavior of electrons in quantum conductors. It has brought new quantities called electronic coherences under the spotlight. In this paper, we explore the relations between electron quantum optics and signal processing through a global review of the various methods for ac- cessing single- and two-electron coherences in electron quantum optics. We interpret electron quantum optics in- terference experiments as analog signal processing, con- verting quantum signals into experimentally observable quantities such as current averages and correlations. This point of view also gives us a procedure to obtain quantum information quantities from electron quantum optics coherences. We illustrate these ideas by discussing two-mode entanglement in electron quantum optics. We also sketch how signal processing ideas may open new perspectives for representing electronic coherences in quantum conductors and understand the properties of the underlying many-body electronic state. Copyright line will be provided by the publisher 1 Introduction Electron quantum optics is a new per- spective on electronic quantum transport that aims at un- derstanding the behavior of electrons in ballistic quantum conductors using paradigms and methods of quantum op- tics [1]. This field has emerged from the development of quantum coherent nanoelectronics during the 90s [2,3,4], the demonstration of electronic interferometers [5,6,7] in quantum Hall edge channels and finally the realization of on-demand single-electron sources [8,9]. Importantly, the introduction of single-electron sources has catalyzed a shift from questions about the statistics of charge flowing across a quantum conductor [10,4] to questions about the wavefunctions of elementary excita- tions carrying the charge. This naturally led to the trans- position of photon quantum optics concepts introduced in the 60s by Glauber to coherent quantum nanoelectronics, thus giving birth to the central concepts of electronic co- herences [11,12,13,14]. Historically, the scattering theory approach to quan- tum transport [15,16,17,18,19] had already emphasized the importance of optics concepts in electronic transport. However, electron quantum optics differs from photon quantum optics because of the Fermi statistics of electrons which changes the nature of the reference “vacuum state” when all sources are switched off. Understanding and ex- tending quantum optics concepts in the presence of such non-trivial vacua was also a motivation for developing electron quantum optics. Even more importantly, elec- trons are charged particles interacting through Coulomb interactions. As stressed out by M. B¨uttiker and his col- laborators [20,21,22,4], this plays a crucial role in high frequency quantum transport by enforcing charge conser- vation and gauge invariance [21,4]. Coulomb interactions also raise the basic question of the fate of electronic quasi- particles in a metal, which was the starting", + "stressed out by M. B¨uttiker and his col- laborators [20,21,22,4], this plays a crucial role in high frequency quantum transport by enforcing charge conser- vation and gauge invariance [21,4]. Coulomb interactions also raise the basic question of the fate of electronic quasi- particles in a metal, which was the starting point for the Landau-Fermi liquid theory [23]. Electron quantum optics thus offers unprecedented possibilities to study these basic condensed matter physics questions down to the single-electron level. These possi- bilities are illustrated by recent two-particle interferometry experiments in quantum Hall edge channels which are reviewed in the present volume [24]. Electron quantum optics also establishes a bridge be- tween quantum coherent nanoelectronics and microwave Copyright line will be provided by the publisher arXiv:1610.02086v3 [cond-mat.mes-hall] 16 Jan 2017 2 B. Roussel et al.: Electron quantum optics as quantum signal processing quantum optics, which now plays an important role in su- perconducting nanocircuits used for quantum information processing and manipulation [25,26,27]. Microwave quan- tum optics is also crucial for understanding the electromag- netic radiation emitted by a quantum conductor, an impor- tant problem rising a growing interest [28,29,30,31,32,33, 34]. The purpose of this paper is to reconsider electron quantum optics from a more global perspective by in- terpreting it in the language of signal processing. In its broadest acceptance, signal processing is an enabling tech- nology that aims at processing, transferring and retrieving information carried in various physical formats called “signals” [35]. Signal processing involves a huge arse- nal of techniques to detect, filter, represent, transmit, and finally extract information or recognize patterns within signals. However, the most famous and historically im- portant examples of signals, such as acoustic or electronic signals, are classical. Here, we would like to emphasize that all electron optics experiments realized so far [36, 37,38,39] as well as proposals for accessing two-electron coherence [14] can be interpreted in the signal processing language as experiments on signals which are no longer classical: namely electron quantum optics coherences. Electronic interferometers realize analog signal processing operations such as “linear filtering” or “overlaps” on these quantum signals and encode the result into experimentally accessible quantities such as average current and current correlations. By emphasizing this point of view, we provide a uni- fied view of the recent developments of electron quantum optics and, as we will show by discussing two-particle in- terferometry, we gain some inspiration towards envisioning new experimental measurement schemes for electronic co- herences. Although this has not really been fully exploited yet, it could also suggest new ways of obtaining data by using suitable sources and data processing for optimizing electronic coherence reconstruction [40]. It may also rise interest in the signal processing community towards elec- tron quantum optics and lead to new innovative experimen- tal and theoretical ideas. In order to develop this point of view, this paper is or- ganized as follows: after briefly recalling the experimen- tal and theoretical context of electron quantum optics in section 2, section 3 reviews the notion of single-electron coherence and the various ways to access this quantity. We will discuss the", + "order to develop this point of view, this paper is or- ganized as follows: after briefly recalling the experimen- tal and theoretical context of electron quantum optics in section 2, section 3 reviews the notion of single-electron coherence and the various ways to access this quantity. We will discuss the signal processing operations performed in various single- and two-particle interferometers used to probe single-electron coherence. Our discussion is com- plementary to that of the review [24] which discusses two- particle interferometry experiments in quantum Hall edge channels and the underlying theory. Here, the same con- cepts are reviewed with a strong emphasis on the quantum signal processing point of view. We then turn to two-electron coherence in section 4, and present its definition and its main properties. We in- troduce its various representations and discuss its non- classical features. We then present its relation to quantum noise of the electrical current, showing the electron quan- tum optics version of Einstein’s relation between particle number fluctuations and wavefunctions. We show that a whole class of experiments can be interpreted as linear filtering converting the intrinsic second order coherence emitted by a source into current correlations. In the last section, we will connect electronic coher- ences to quantum information theoretical quantities. For this purpose, we will explain how to derive effective qubit density matrices from a set of orthogonal single-particle states. In particular, we will illustrate this by discussing electron/hole entanglement in the many-body state gener- ated by a mesoscopic capacitor. We will finally sketch how ideas from signal processing lead to a suitable definition of these density matrices for periodically driven systems. 2 The context of electron quantum optics Let us briefly review the main steps that have lead to the develop- ment of electron quantum optics. 2.1 Experiments On the experimental side, the in- teger quantum Hall effect in high mobility 2-dimensional electron gas (2DEG) in AsGa/AsGaAl heterostructures provides the analogous of optical fibers through chiral propagation of charge carriers within the so-called edge channels. Progresses in nanofabrication not only enabled the fabrication of the quantum point contact (QPC), which plays the role of an ideal electronic beam splitter [41,42, 43], but have also enabled its embedding into complicated geometries such as the electronic Mach-Zehnder interfer- ometer (MZI) [5,44,45] and the Samuelsson-B¨uttiker in- terferometer [46,7]. These pioneering experiments showed that, at low temperatures, electronic coherence can be maintained over distances comparable to the size of these circuits (from few to 20 µm) [47]. They have also opened the way to the demonstration and study of more complex electronic circuits in which elements such as QPC, quan- tum dots and single-electron sources could be placed like optical components on an optical table. As mentioned in the Introduction, the demonstration of the mesoscopic capacitor as an on-demand single-electron source [8] marked the beginning of electron quantum optics. Now, other single-electron sources have been demonstrated in AsGa/AsGaAl, from turnstiles [48,49], mainly motivated by metrology, to the Leviton source [9] which is reviewed in this volume with great details [50]. Around the same time, progresses in", + "as an on-demand single-electron source [8] marked the beginning of electron quantum optics. Now, other single-electron sources have been demonstrated in AsGa/AsGaAl, from turnstiles [48,49], mainly motivated by metrology, to the Leviton source [9] which is reviewed in this volume with great details [50]. Around the same time, progresses in microwave tech- nology and measurement techniques have lead to the ex- ploration of high-frequency transport [51,52], thereby confirming the predictions by B¨uttiker and his collabo- rators [53,21]. From a broader perspective, these devel- opments have opened the way to in-depth experimental investigations of high-frequency quantum coherent nano- electronics. Copyright line will be provided by the publisher pss header will be provided by the publisher 3 2.2 Theory The electron quantum optics formalism was then developped taking advantage of the chirality of electronic transport in quantum Hall systems: in such sys- tems, the optical analogy is exploited at its best by decom- posing the quantum conductor, or more generally the elec- tronic circuit, into simple building blocks such as quan- tum point contacts or energy filters so that an incoming electronic flow is transformed into an outgoing one. Then, within the measurement stage, the average current or the current noise is measured either at zero or at a given high frequency. The resulting formalism bears a close similar- ity with the input-output formalism of photon quantum op- tics [54]. In particular, it assumes that electronic coher- ences are probed within a region where interactions can be neglected. In such a region, electrons propagate freely at a Fermi velocity which will be denoted by vF throughout the present paper. When electronic coherences are probed at a given position, often corresponding to the position of a detector, they depend on time variables. Such a description contains an assumption on the screening of Coulomb interactions. For example, we as- sume that screening at a quantum point contact is good enough to neglect any capacitive coupling between the in- coming and outgoing edge channels. As of today, there has been no experimental evidence contradicting this assump- tion. Within this framework, interaction effects have been studied extensively, starting with MZI. We will very brielfy discuss some of these works in section 3.2 mentioning that interaction effects can lead to a breaking of the paradigm of quantum conductors as linear electron quantum optics components. We also refer the reader interested by inter- action induced decoherence effects in Hong Ou Mandel (HOM) experiments to [24] as well as [39]. 2.3 New systems The rapid development of electron quantum optics has also catalyzed a stream of works whose purpose is to extend its application range to new physical systems. A first line of research deals with extending electron quantum optics to situations in which interactions poten- tially lead to a drastic change of the ground state such as in superconductivity. This has led to study of electron quan- tum optics with Bogoliubov quasi-particles which is re- viewed in this volume [55]. Another question is the gener- alization of electron quantum optics to fractional quantum Hall (FQH) edge channels. Proposals have been", + "change of the ground state such as in superconductivity. This has led to study of electron quan- tum optics with Bogoliubov quasi-particles which is re- viewed in this volume [55]. Another question is the gener- alization of electron quantum optics to fractional quantum Hall (FQH) edge channels. Proposals have been made for single quasi-particle and single electron emitted in these systems [56] and the HOM experiment with Lorentzian pulses has been considered [57]. However, a full general- ization of electron quantum optics in the FQH regime is still missing, the main problem being the absence of any ideal quasi-particle beam splitter. Nevertheless, perturba- tive approaches may prove to be useful for experiments whenever the crossover energy scale associated with a con- striction [58] is well below the experimentally relevant en- ergy scales. Another line of research focuses on manipulating the spin degree of freedom at the single-electron level. Quan- tum spintronics has risen a strong interest in the meso- scopic physics community because of the importance of coherent spin transport and manipulation for quantum in- formation processing. Although the ν = 2 edge channel system had already been envisioned for quantum spintron- ics [59], 2D topological insulators (TI), such as quantum spin Hall systems [60], are now considered as a poten- tially important class of systems for electron quantum op- tics. In these materials, such as CdTe/HgTe and InAs/GaSb quantum wells at zero magnetic field, edge channels are topologically protected from backscattering. They come as counterpropagating pairs with opposite spin polariza- tion (spin-momentum locking). The mesoscopic capacitor built from a 2D TI has been proposed as an on-demand single Kramer pair source emitting two single-electron ex- citations with opposite spins [61,62]. The HOM experi- ment has been discussed with such sources [63]. A variant of this source based on a driven antidot has also recently been proposed [64] as well as a different system relying on two quantum dots coupled to a 2D TI via tunneling barriers [65]. Coulomb interactions are expected to play an impor- tant role in these systems due to the spatial superposition of two counter-propagating edge channels. Comparing to the ν = 2 quantum Hall edge channel system, interactions among counterpropagating edge systems are expected to induce new effects ranging from resonance effects [66] to fractionalization [67]. Although these developments go beyond the scope of the present paper, it is important to keep in mind that the basic concepts of electron quantum optics can be extended to quantum spintronics in a relatively simple way. 3 Single-electron coherence Let us now review the concept of single-electron coherence and discuss how it can be accessed through single-particle interferometry and two-particle interferometry. Our main message is that, un- der certain circumstances, the first type of experiments cor- respond to linear filtering in the signal processing language whereas the HOM experiment performs an analog compu- tation of the overlap of two single-electron coherences. 3.1 Definition and representations Single-electron coherence [11] is defined by analogy with first-order coher- ence in photon quantum optics [68]: G(e) ρ (1|1′) = Tr(ψ(1) ρ ψ†(1′)) (1) where", + "in the signal processing language whereas the HOM experiment performs an analog compu- tation of the overlap of two single-electron coherences. 3.1 Definition and representations Single-electron coherence [11] is defined by analogy with first-order coher- ence in photon quantum optics [68]: G(e) ρ (1|1′) = Tr(ψ(1) ρ ψ†(1′)) (1) where ρ denotes the reduced density operator for the elec- tronic fluid, ψ and ψ† denote the fermionic destruction and creation field operators and 1 = (α, x, t) and 1′ = (α′, x′, t′) denote edge channel indices and space time co- ordinates. For spin polarized edge channels, α and α′ cor- respond to spin indices: α = α′ then encodes spin popu- lations whereas α = −α′ corresponds to spin coherences, thus showing that the formalism of electron quantum optics can be easily extended to account for the spin. Copyright line will be provided by the publisher 4 B. Roussel et al.: Electron quantum optics as quantum signal processing In the following, to simplify notations, channel/spin in- dices will be dropped out when only one edge channel is involved. single-electron coherence contains all the in- formation on single-electron wavefunctions present in the system. For example, let us consider the N electron state |ΨN⟩= QN k=1 ψ†[ϕk]|∅⟩where the creation operator for an electron in the single-particle state ϕk is defined by: ψ†[ϕk] = vF Z R ϕk(t)ψ†(t) dt . (2) This state is obtained by filling mutually orthogonal single- particle states (ϕk)k∈{1,...,N} on top of the electronic vac- uum |∅⟩. Then, a straightforward application of Wick’s the- orem shows that, in the space domain at the initial time: G(e) |ΨN⟩(x|y) = N X k=1 ϕk(x) ϕk(y)∗. (3) In a metallic conductor, at zero temperature and with all electronic sources switched off, the reference state is a Fermi sea vacuum of given chemical potential µ(x). Therefore, contrary to the case of photon quantum op- tics, the single-particle coherence does not vanish when sources are switched off but reduces to the Fermi sea con- tribution G(e) Fµ(x)(t|t′) = ⟨Fµ(x)|ψ†(x, t′)ψ(x, t)|Fµ(x)⟩. On the contrary, the inter-channel single-electron coher- ence vanishes when all electronic sources are switched off: G(e) F (α, t|α′, t′) = 0 for α ̸= α′. In electron quantum optics, G(e) ρ is considered at a given position x within the electronic circuit, thus lead- ing to a two-time function G(e) ρ,x(t|t′) = G(e) ρ (x, t|x, t′). When the sources are switched on, the single-electron co- herence is different from the Fermi sea contribution and the excess single-electron coherence is defined by subtract- ing the Fermi sea contribution: ∆G(e) ρ,x(t|t′) = G(e) ρ,x(t|t′) − G(e) Fµ(x)(t|t′). The most convenient representation for single-electron coherence is a mixed time-frequency representation called the electronic Wigner distribution function, which captures both temporal evolution and the nature of excitations [40]: W (e) ρ,x(t, ω) = Z +∞ −∞ vF G(e) ρ,x \u0010 t + τ 2 t −τ 2 \u0011 eiωτdτ (4) where vF denotes the Fermi velocity at position x. The electronic Wigner distribution function is real. Its marginal distributions give access to the time-dependent", + "nature of excitations [40]: W (e) ρ,x(t, ω) = Z +∞ −∞ vF G(e) ρ,x \u0010 t + τ 2 t −τ 2 \u0011 eiωτdτ (4) where vF denotes the Fermi velocity at position x. The electronic Wigner distribution function is real. Its marginal distributions give access to the time-dependent average electric current and to the electronic distribution function fe(ω) at position x: fe(ω) = W (e) ρ,x(t, ω) t (5a) ⟨i(x, t)⟩ρ = −e Z +∞ −∞ ∆W (e) ρ,x(t, ω) dω 2π (5b) A classicality criterion for the electronic Wigner distribu- tion function has been formulated [40]: 0 ≤W (e) ρ,x(t, ω) ≤ 1. When verified, it basically means that W (e) ρ,x(t, ω) can be interpreted as a time-dependent electronic distribution function. This is the case for a quasi-classically driven Ohmic contact, when |eVac| ≫hf and kBT ≫hf, f be- ing the driving frequency. In this case, both thermal fluc- tuations and the ac drive are responsible for multiphoton transitions in the electronic fluid and many electron/hole pairs are generated. On the contrary, for a single-electron excitation, quantum delocalization is responsible for neg- ativities as seen on the example of the MZI [40]. Access- ing single-electron coherence would thus enable to discuss the non classicality of the electronic fluid at single-particle level. 3.2 Single-particle interferometry as linear filter- ing Introduction In classical signal processing, linear fil- ters transform time-dependent input signals into output sig- nals under the constraint of linearity. Well known examples include linear circuit elements in classical electronics and linear optics elements such as lenses, beam splitters and other various optical devices. These components act as lin- ear filters on the electromagnetic field classical coherence introduced in the 30s [69]. This statement also extends to quantum optics by considering quantum optical coherences introduced by Glauber [68]. In this section, we show that the same statement is true in electron quantum optics provided we use quantum con- ductors in which interaction effects can be neglected. As an example, we will see how the ideal Mach-Zenhder in- terferometer [70] or the measurement of the electronic dis- tribution function using a quantum dot as an energy fil- ter [71] realize linear filtering of the excess single-electron coherence ∆G(e) ρ,x(t|t′), which should therefore be seen as a “quantum signal” depending on two times. We will then discuss how Coulomb interactions partly – but not totally – invalidate this statement. In particular, we will explain why measuring finite-frequency currents enables to keep track of interaction effects and to recover information on single-electron coherence. Mach-Zehnder interferometry An ideal electronic Mach-Zehnder interferometer, such as the one depicted on Fig. 1, is characterized by the time of flights τ1,2 along its two arms and the magnetic flux threading it ΦB = φB × (h/e). When an electronic source S is placed on the incoming edge channel 1, the time-dependent outgo- ing electric current in channel 1 is directly proportional to the excess electronic coherence of the source [70,40]: ⟨i1out(t)⟩= X j=1,2 Mj,j⟨iS(t −τj)⟩ (6a) −2e|M1,2| Z R cos (ωτ12 + φ) ∆W (e) S (t", + "an electronic source S is placed on the incoming edge channel 1, the time-dependent outgo- ing electric current in channel 1 is directly proportional to the excess electronic coherence of the source [70,40]: ⟨i1out(t)⟩= X j=1,2 Mj,j⟨iS(t −τj)⟩ (6a) −2e|M1,2| Z R cos (ωτ12 + φ) ∆W (e) S (t −¯τ, ω) dω 2π (6b) Copyright line will be provided by the publisher pss header will be provided by the publisher 5 A B 1in 1out 1MZI τ1 2in 2out 2MZI τ2 ΦB Figure 1 Schematic view of the Mach-Zehnder interfer- ometer: the incoming channels are partitionned at the elec- tronic beam splitter A and then recombined by the beam splitter B. Here τ1 and τ2 denote the times of flight across the two branches of the MZI and ΦB the magnetic flux en- closed by the interferometer. where Mi,j denotes the product AiA∗ j, Aj being the trans- mission amplitude of the beam splitters along path j of the interferometer. We have introduced τ12 = τ1 −τ2, ¯τ = (τ1 + τ2)/2 and φ = arg(M1,2 + 2πφB) which is the phase associated with both beam splitters and the mag- netic flux. The first line (Eq. (6a)) does not depend on the magnetic flux and therefore corresponds to classical prop- agation within each of the two arms of the MZI, whereas the second line (Eq. (6b)) corresponds to quantum inter- ferences between propagations within both arms. Because the average electric current is also proportional to the ex- cess single-electron coherence of the source, the outgoing average current is obtained from the excess incoming co- herence ∆G(e) S in channel 1in by a linear filter which we write symbolically ⟨iout,dc⟩= LMZI[∆G(e) S ]. Measurements of the ΦB dependent part of the average dc current for vari- ous τ1−τ2 could then be used to reconstruct single-electron coherence [70]. The key ingredient in this derivation is the absence of electronic interactions. Whenever one replaces the Mach- Zehnder interferometer by an ideal ballistic quantum con- ductor in which interactions are neglected, the outgoing current in the measurement lead would also be proportional to ∆G(e) S . Denoting by S(tf, ti) the scattering amplitude for an electron arriving into the conductor at time ti and going out towards the measurement lead at time tf, then the outgoing average time-dependent current is given by ⟨iout(t)⟩= Z R2 S(t, t+) S∗(t, t−)∆G(e) S (t+, t−) dt+dt− (7) which describes a linear filtering of the incoming single- electron coherence ∆G(e) S associated with time-dependent scattering. In particular, this expression is valid within the framework of Floquet scattering theory [72]. On the role of interactions However, as we shall now discuss, because of Coulomb interactions, the situa- tion in electron quantum optics is subtler than in photon quantum optics. It is therefore important to clarify in which Interaction region environment (in) |0⟩ edge channel (in) N ω |Λω⟩ environment (out) N ω |r(ω)Λω⟩ edge channel (out) N ω |t(ω)Λω⟩ Figure 2 Input/output treatment of a finite-length interac- tion region (light grey box) in the edge-magnetoplasmon scattering framework: an edge channel", + "is therefore important to clarify in which Interaction region environment (in) |0⟩ edge channel (in) N ω |Λω⟩ environment (out) N ω |r(ω)Λω⟩ edge channel (out) N ω |t(ω)Λω⟩ Figure 2 Input/output treatment of a finite-length interac- tion region (light grey box) in the edge-magnetoplasmon scattering framework: an edge channel (in red) is ca- pacitively coupled to another conductor in the linear response regime, described by an environmental chan- nel (in blue). At zero temperature, the incoming en- vironmental modes are in their vacuum state. When a coherent edge-magnetoplasmon state |Λ⟩ = ⊗ω |Λω⟩ is sent in the incoming channel, the outgoing state is ⊗ω (|t(ω)Λω⟩⊗|r(ω)Λω⟩) where t(ω) denotes the edge- magnetoplasmon transmission amplitudes across the inter- action region and r(ω) the amplitude to be scattered into enviromental modes. situations the linear filtering paradigm is valid and when it is not valid. To begin with, let us consider a simple situation in which the source is placed at the input of a finite-length interaction region, as depicted on Fig. 2. For simplicity, we assume that this interaction region can be described within the edge-magnetoplasmon scattering formalism [73,74]: the incoming edge-magnetoplasmon mode at a given fre- quency is scattered elastically between the corresponding outgoing mode and environmental modes associated with other electrical degrees of freedom present in the problem. The environment can involve edge channels or any neigh- boring linear circuit modeled as a transmission line with a frequency-dependent impedance [75]. Such a modeliza- tion is valid as long as all the components of the system are in the linear response regime. This formalism was used to compute the effect of Coulomb interactions on coher- ent current pulses [76] and single-electron excitations [74, 77]. Equivalently, because they induce inelastic processes, Coulomb interactions turn a quantum conductor into a non- linear component from the electron quantum optics point of view. Consequently, in general, the excess outgoing sin- gle electron coherence is not a linear filtering of the incom- ing one! A criterion for the validity of the electronic scatter- ing theory approach to quantum transport at finite fre- quencies is that the frequency dependence of the elec- tronic scattering matrix of a quantum conductor can be ne- glected [4]. Single- to few-electron excitations emitted by Copyright line will be provided by the publisher 6 B. Roussel et al.: Electron quantum optics as quantum signal processing electron quantum optics sources such as the mesoscopic capacitor [8] or the Leviton source [9] as well as periodic electric currents generated using an advanced waveform generator usually define a frequency scale in the range of one to few tens of GHz. On the other hand, an extended conductor such as a MZI has a scattering matrix varying over frequency scales of the order of the inverse of the time of flight of the conductor. For a 10 µm interferometer, it is of the order of 10 GHz. This is why extended quantum con- ductors such as MZIs fail to satisfy this criterion. The im- portant stream of theoretical work on interaction-induced decoherence [78,79,80,81,82] in MZI interferometers il- lustrate", + "time of flight of the conductor. For a 10 µm interferometer, it is of the order of 10 GHz. This is why extended quantum con- ductors such as MZIs fail to satisfy this criterion. The im- portant stream of theoretical work on interaction-induced decoherence [78,79,80,81,82] in MZI interferometers il- lustrate the full complexity of understanding interaction ef- fects in such extended quantum conductors. More recent works [83,84] dealing with the propagation of individual- ized energy-resolved single-electron excitations in a MZI are directly relevant for electron quantum optics but also show that this problem is not yet fully understood even at the single-electron level. By contrast, Coulomb interaction effects can be ne- glected over a much broader frequency range in the QPC which is an almost point-like electronic beam-splitter. As we shall see in the next section, this plays a very important role for the HOM and HBT experiments. Last but not least, the average finite-frequency currents are relatively robust to the effect of Coulomb interactions: whenever all electrical components respond linearily to the incoming excitation, the edge-magnetoplasmon scattering matrix can be used to reconstruct the incoming average fi- nite frequency currents from the outgoing ones. Measure- ments of finite-frequency average currents have been suc- cessfully developped in the 1-11 GHz range to perform the first direct measurement of edge-magnetoplasmon scatter- ing amplitudes [85]. 3.3 Two-particle interferometry as overlap Motivation Although amplitude interferometry relies on the measurement of average currents, it does not seem well suited to perform single-electron tomography. First of all, as in optics, it would require a perfect control on elec- tronic optical paths down to the Fermi wavelength. More importantly, as discussed in the previous section, Coulomb interactions prevent reconstructing the incoming single- electron coherence from the experimental signals. This situation is very similar to what happened in as- tronomy in the 30s and 40s: attempts at directly measuring the diameter of normal stars using amplitude interferom- etry were plagued by atmospheric turbulences and by the technological challenge of building a large optical interfer- ometer. However, a way to circumvent this bottleneck was found by Hanbury Brown and Twiss (HBT) in the 50s [86]: their idea was to measure intensity correlations [87] since these contain interferences between waves emitted by pairs of atoms on the star. In quantum optics, the HBT effect ultimately relies on two-photon interferences [88]. In the 80s, the Hong Ou Mandel (HOM) experiment [89] also demonstrated two-particle interferences for identical par- ticles (photons). Since then, two-particle interference ef- fects have been observed in many different contexts, from stellar interferometry to nuclear and particle physics [90] and more recently with bosonic as well as fermionic cold atoms [91]. Recent experiments demonstrate a higher de- gree of control by using independent single-photon [92] and single atom sources [93]. In this section, we review how the HOM experiment can be used to measure the overlap of the excess single- particle coherences arriving at a beam splitter. Remark- ably, this result is true not only for electrons but for any fermionic or bosonic excitation. In photon quantum optics, it", + "[93]. In this section, we review how the HOM experiment can be used to measure the overlap of the excess single- particle coherences arriving at a beam splitter. Remark- ably, this result is true not only for electrons but for any fermionic or bosonic excitation. In photon quantum optics, it forms the basis of homodyne tomography [94,95] re- cently used to characterize few-photon states in the optical domain [96]. In the microwave domain, the HOM scheme has been used to access photon quantum optical correla- tions from electrical measurements [27,26] and forms the basis of a tomography scheme for itinerant microwave pho- tons [97,25]. Theoretical analysis In electron quantum optics, the HBT and HOM experiments are demonstrated by sending electronic excitations generated by one or two electronic sources on an ideal electronic beam splitter, as depicted on Fig. 3. In order to make a precise analogy with photon quan- tum optics, keeping in mind that in electron quantum op- tics, the vacuum is the reference Fermi sea and not a true vacuum, we consider that the electronic analogue of the table-top HBT experiment (Fig. 3a) [87] is realized when one of the incoming channels is fed with the reference Fermi sea (S1 or S2 being off). By the same analogy, the electronic HOM experiment (Fig. 3b) corresponds to situ- ations with both electronic sources in the incoming chan- nels switched on. Finally, whereas in photon quantum op- tics, the arrival of individual photons can be recorded, in electronics, the quantities of interest are the current corre- lations at zero frequency in the two outgoing branches. Let us focus on the outgoing current noise in chan- nel 1. A first important point is that the low-frequency cur- rent noise does not depend on the distance to the QPC: in- teraction effects lead to edge-magnetoplasmon scattering among the various outgoing edge channels close to the one considered but the total power remains the same. This is why HBT/HOM interferometry is immune to interaction effects in the measurement stage (beyond the QPC). In the same way, the intensity correlations measured in an optical stellar HBT interferometer are not blurred by atmospheric turbulences. Consequently, what we need is the excess low fre- quency current noise just after the QPC when both sources S1 and S2 are switched on. It is the sum of three contribu- tions [40]: ∆S(S1&S2) 11 = ∆S(S1) 11 + ∆S(S2) 11 + ∆S(HOM) 11 (8) where ∆S(S1) 11 and ∆S(S2) 11 are the excess current noise when only the source S1 (resp. S2) is switched on. These Copyright line will be provided by the publisher pss header will be provided by the publisher 7 source S BS 1out 2out correlator mobile a) The Hanbury Brown Twiss experiment S1 S2 BS 1out 2out correlator b) The Hong Ou Mandel experiment Figure 3 Principle of the HBT and HOM experiments: in the optical HBT experiment (a), excitations emitted by a single source S are partitioned at the beamsplitter BS whereas in the HOM experiment (b), excitations emitted by two sources S1 and S2 are", + "correlator b) The Hong Ou Mandel experiment Figure 3 Principle of the HBT and HOM experiments: in the optical HBT experiment (a), excitations emitted by a single source S are partitioned at the beamsplitter BS whereas in the HOM experiment (b), excitations emitted by two sources S1 and S2 are sent onto BS. In optics, one performs a time-resolved detection of photons. In the electronic case, the beamsplitter is a QPC, and one mea- sures current correlations between 1out and 2out or cur- rent noise in the 1out channel. In the case of the HBT ex- periment, vacuum is replaced by the reference Fermi sea. contributions correspond to the excess noise in HBT exper- iments performed on each of the sources. The last term ∆S(HOM) 11 = −2e2RT Z R2 ∆W (e) 1in(t, ω)∆W (e) 2in(t, ω) dt dω 2π (9) is the overlap of the excess single-electron coherences ar- riving at the QPC [40] (R and T denoting the reflection and transmission probabilities of the QPC). Eq. (9) encodes the effect of two-particle interferences between the excitations emitted by these sources. Note that the time delay of the two sources can be controlled and therefore a single exper- imental run gives access to the time-shifted overlaps of the excess electronic Wigner functions of the two sources. Fi- nally, the minus sign comes from the fermionic statistics of electrons. The two first terms also involve two-particle quantum interferences. Because they contain information on coher- ences of each of the sources, they will be discussed in section 4. Our point here is to emphasize that the elec- tronic HOM experiment automatically encodes into the ex- perimental signal what the signal processing community would call the sliding inner product of the quantum signals formed by the incoming excess single-electron coherences in channels 1 and 2. This is why the HOM experiment is so important: it can be used to test for unknown excess elec- tronic Wigner functions by looking at their overlaps with themselves or with the ones generated by controlled and calibrated sources. This idea has been expanded to describe a generic tomography protocol for reconstructing an un- known excess single-electron coherence from its overlaps with coherences generated by suitable ac + dc drives [98]. We refer the reader to [24] in the same volume for a de- tailed description of this protocol. Experimental demonstration The electronic HBT experiment has been demonstrated in the late 90s using dc sources [3,2] and more recently using single-electron sources [99] which were then used to perform the elec- tronic HOM experiment [36]. These experiments have paved the way to measurements and studies of electron de- coherence down to the single-electron level through HOM interferometry. The idea of the tomography protocol has been re- cently demonstrated by D.C. Glattli’s group [37]: in this experiment, a stream of Lorentzian pulses is sent onto a beam splitter whose other incoming channel is fed with a small ac drive on top of a dc bias. Measurement of the low-frequency noise then enables reconstructing the photo-assisted transition amplitudes which, in this", + "D.C. Glattli’s group [37]: in this experiment, a stream of Lorentzian pulses is sent onto a beam splitter whose other incoming channel is fed with a small ac drive on top of a dc bias. Measurement of the low-frequency noise then enables reconstructing the photo-assisted transition amplitudes which, in this case, contain all the information on single-electron and higher- order electronic coherences [9]. This experiment being performed in a 2DEG at zero magnetic field, interaction effects can be neglected and the experiment leads to the re- construction of the Leviton single-electron coherence [37]. As reviewed in this volume [24], the HOM exper- iment has also recently been used to probe interaction effects within quantum Hall edge channels. In these ex- periments, performed at filling factor ν = 2, two single- electron sources are located at some distance of the QPC and interaction effects are strong enough to lead to quasi- particle destruction, as suggested by energy relaxation experiments [100]. First, the HOM effect was used to probe how interactions lead to fractionalization of classi- cal current pulses [38] in qualitative agreement with the neutral/charge edge-magnetoplasmon mode model [80] which had been already probed through energy relaxation experiments [101] and high-frequency admittance mea- surements [85]. But the real strength of HOM experiment comes from its ability to probe electronic coherence in a time- and energy-resolved way. It was thus recently used to study quantitatively the effect of Coulomb interactions on energy-resolved single-electron excitations (called Landau Copyright line will be provided by the publisher 8 B. Roussel et al.: Electron quantum optics as quantum signal processing excitations) [39]. The experimental data confirm theo- retical predictions and validate the decoherence scenario based on edge-magnetoplasmon scattering [102,77]. 4 Two-electron coherence Let us now turn to two- electron coherence. After briefly reviewing its definition and properties, introducing the two-electron Wigner dis- tribution function will enable us to emphasize its non- classical features arising from Fermi statistics. We will then turn to two-particle interferometry and show that, un- der appropriate hypotheses, these experiments perform a linear filtering on the intrinsic two-electron coherence, thus generalizing what was discussed in section 3.2. 4.1 Definition and representations Definition Two-electron coherence is defined by di- rect analogy with Glauber’s second-order photonic coher- ence [13]: G(2e) ρ (1, 2|1′2′) = Tr(ψ(2)ψ(1)ρ ψ†(1′)ψ†(2′)) (10) where 1 = (α1, x1, t1) and 2 = (α2, x2, t2), α1,2 being channel indices, and similarly for 1′ and 2′. Its physical meaning can be obtained by computing the two-electron coherence for the state |ΨN⟩defined in section 3.1. The result is a sum over all the two-electron wavefunctions ϕk,l(x, y) = ϕk(x)ϕl(y) −ϕk(y)ϕl(x) (k < l) that can be extracted from the N single-particle wavefunctions (ϕk)k∈{1,...,N} [14]: G(2e) |ΨN⟩(x1, x2|x′ 1, x′ 2) = X k mu. They have been sought assiduously, but have not yet been found; however at a recent conference Minamisono et al. [2] reported a preliminary sighting of such currents in the beta decays of mass 12 nuclei at a level of a few %. It would be interesting to compare any findings with theoretical work as a test of the standard electroweak theory; to my knowledge, no such detailed calculations have yet been done. Another use of electrons emitted in", + "the beta decays of mass 12 nuclei at a level of a few %. It would be interesting to compare any findings with theoretical work as a test of the standard electroweak theory; to my knowledge, no such detailed calculations have yet been done. Another use of electrons emitted in superallowed Fermi beta decays (0+ −→0+) in an isomultiplet is as a test of the standard model via unitarity of the CKM matrix.1 Could there be a missing interaction, f.i.? The matrix element Vud connecting up and down quarks is by far the largest one in the unitarity of U ≡| Vud |2 + | Vus |2 + | Vub |2= 1 Here Vud and Vub connect the up quark with the strange and bottom quarks, respectively. The precise measurements of superallowed transitions together with radiative corrections and removal of charge dependent nuclear effects allow one to determine Vud to better than 10−3. In addition, these measurements, (Fig.1), show that CVC holds to ∼4 × 10−4. Fig.1. Ft values for 0+ →0+β–decays. A straightforward analysis of the experiments, including a recent 10C experiment [3], gives Vud = .9740 ± .0006. Together with the measurements of Vus and Vub, one then obtains U = .9972 ± .0019. However, it has been proposed that the nuclear charge–dependent corrections should be corrected by a smooth Z-dependence. [4] In that case U = .9980±.0019. Is the discrepancy from unitarity meaningful? Personally, I doubt it. The largest uncertainty may reside in the charge–dependent nuclear corrections and there is now an attempt to push the calculation of these corrections to increased accuracy. [5] Lastly let me mention tests of time reversal invariance. [6] I will not describe the standard searches for TRI–odd terms (D ⃗J · ⃗pe × ⃗pν) in beta decays, particularly 19Ne, where ⃗J is the spin direction of the parent nucleus. Recently there have been searches for the R–term, R ⃗J ·⃗j ×⃗pe in the beta decays of 8B and 8Li, where ⃗j is the spin direction of the electron; the advantage of these nuclei is that they decay to the unstable 8Be, which breaks up into two alpha particles, which readily can be detected. This experiment is underway. [7] Another 2 method has been used at the Sherrer Institute by Allet et al; [8] they measure the transverse polarization of the emitted electrons directly and thus find a limit ImCT/CA < .012 at the 68% level; here CT and CA are the coefficients of the axial tensor and axial vector currents. It would be useful if these experiments could be increased in accuracy so as to at least reach the level where final state interactions spoil TRI tests; here this would be 7 × 10−4. The experimenters still have a ways to go. B. Double Beta Decay [9] The decay (A, Z) −→(A, Z + 2) + e−+ e−+ ¯ν + ¯ν is expected in the standard model and has been seen seen in several nuclei (82Se, 100Mo, and 150Nd) with half lives of about 1020y, consistent with the standard model. [10] Searches for", + "Double Beta Decay [9] The decay (A, Z) −→(A, Z + 2) + e−+ e−+ ¯ν + ¯ν is expected in the standard model and has been seen seen in several nuclei (82Se, 100Mo, and 150Nd) with half lives of about 1020y, consistent with the standard model. [10] Searches for the no neutrino decay mode, important for determining whether ν’s are massive and of the Majorana type, have been continuosuly improved; they use Ge crystals; at present the lower limit on the half life is 3 × 1024y for 74Ge. This experiment sets a lower bound on the mass of the most massive Majorana neutrino mass, mν r 1024y t1/2(76Ge). Improved experiments hope to push this mass down further by one to two orders of magnitude or find the double beta decay. C. Parity Nonconservation Studies [11] The first test of the electroweak theory and its neutral currents came from electron scattering on hydrogen and deuterium at SLAC. At lower energies, parity violating (pv) scattering of electrons on nucleons and nuclei allows one to determine all four weak currents of the nucleons: the axial vector and the vector couplings to protons and neutrons. The experiments are difficult and so far only one experiment of polarized electrons on 12C has been carried out at MIT. [12] The results agree with the theory at a level of 10%. More recently, in an ongoing experiment (SAMPLE) at MIT, the weak interaction of the electron and proton is being used to investigate strangeness in the nucleon. As in all pv experiments, it is the interference of the weak interaction with the electromagnetic one that is being detected by searching for a parity-odd signal such as < ⃗j > ·⃗p, where ⃗p is the incident momentum of the electron and < ⃗j > is its polarization. Initial evidence for non-vanishing strangeness matrix elements in the nucleon came from measurements of the spin structure of the proton by polarized electrons on polarized nuclear targets. This indicated that △s ≡s ↑−s ↓+¯s ↑−¯s ↓≈0.1 −0.2. (1) Further indications of a non-negligible fraction of strangeness came from elastic neutrino scattering on protons. At the present time, the axial vector strangeness matrix element for the proton stands at 0.1 ± .03. For elastic e−p scattering, the standard model gives Jem µ = ¯u(p′)[γµF em 1 + iσµν qν 2M F em 2 ]u(p), (2) 3 JZ µ = ¯u(p′)[γµF Z 1 + iσµν qν 2M F Z 2 + γµF Z A γ5]u(p), (3) If strangeness is included, then SU(3) notation is helpful, and we have F em 1 = 1 2 h F (8) 1 + τ3F (3) 1 i =⇒1 for p, (4) F em 2 = 1 2 h (κp + κn)F (8) 2 + τ3(κp −κn)F (3) 2 i =⇒κp for p, (5) F Z 1 = 1 2 h −F (0) 1 + τ3yF (3) 1 + yF (8) 1 i =⇒1 2 \u0010 1 −4 sin2 θW \u0011 for p, (6) F Z 2 = 1 2[−g2F (0) 2 + τ3y(κp −κn)F (3) 2 +", + "2 i =⇒κp for p, (5) F Z 1 = 1 2 h −F (0) 1 + τ3yF (3) 1 + yF (8) 1 i =⇒1 2 \u0010 1 −4 sin2 θW \u0011 for p, (6) F Z 2 = 1 2[−g2F (0) 2 + τ3y(κp −κn)F (3) 2 + y(κp + κn)F (8) 2 ] =⇒1 2 h\u0010 1 −4 sin2 θW \u0011 κp −κn −κs i for p, (7) F Z A = 1 2 h −g(0) A F (0) A + 2gAτ3F (3) A + (6F −2D)F (8) A i =⇒−1 2g(0) A + gA + (3F −D) for p, (8) where form factors have been normalized to unity at squared momentum transfers Q2 = 0, and the proton values are given at Q2=0. We also use y = \u0010 1 −2 sin2 θW \u0011 , gA = 1.26, g2 = κp + κn + κs, 6F −2D ≃1.1 where F and D are the fractions that are odd and even, respec- tively under SU(3); κp (κn) is the anomalous magnetic moment of the proton (neutron), κs is the strange magnetic moment, and the superscripts on the form factors refer to SU(3) transformation properties. There are two new and unconstrained couplings and form factors, g2F (0) 2 and g(0) A F (0) A , where F s i = F (0) i + F (8) i . The Sample experiment at MIT has as its aim the measurement of g2F (0) 2 (Q2) at small Q2 where F (0) 2 ≈1. The experimenters measure the cross section of 200 MeV electrons polarized parallel and antiparallel to their momenta on a H target. The asymmetry a is given by a = dσR −dσL dσR + dσL = −GQ2 √ 2πα{[2τ tan2 θ 2(F em 1 + F em 2 )(F Z 1 + F Z 2 ) + F em 1 F Z 2 + F em 2 F Z 2 τ] −E + E′ 2M tan2 θ 2(1 −4 sin2 θW)F Z A (F em 1 + F em 2 )} ×{(F em 1 )2 + τ(F em 2 )2 + 2τ tan2 θ 2(F em 1 + F em 2 )2}−1 (9) where E(E′) is the initial (final) electron energy, θ is the scattering angle, and τ ≡Q2/4M2. The last term in Eq.(9) is small because (1−4 sin2 θW) ≈0.1, and the second one is small at back angles, where the first term dominates. The experiment is carried out for an average 4 Q2 ≈0.1GeV 2(130◦< θ < 170◦). At these angles, the asymmetry is sensitive to F Z 2 and thus allows a determination of κs. The results to date [13] are shown in Fig.2, Fig.2. Results for the parity–violating asymmetry measured in the 1995 and 1996 running periods. The hatched region is the asymmetry band (due to the axial radiative correction) for µs = Gs M = 0. 5 where the hatched band corresponds to κs = 0 (it is a band due to uncertain radiative corrections). At Q2 = 0.1 GeV2, the authors find κsF", + "running periods. The hatched region is the asymmetry band (due to the axial radiative correction) for µs = Gs M = 0. 5 where the hatched band corresponds to κs = 0 (it is a band due to uncertain radiative corrections). At Q2 = 0.1 GeV2, the authors find κsF s = 0.23 ± 0.37 ± 0.15 ± 0.19n.m., where the first error is statistical, the second one is systematic and the third one is due to axial radiative corrections. At present, κs is consistent with zero, but its possible positive value is opposite to that predicted by most theorists [14], e.g. by using N →ΛK →N. The SAMPLE experiment will measure the coupling of the Z-boson to the proton in the future; also work at MIT and at TJNAF is expected to bring the errors down to a level of ±.1 n.m. (nucleon magneton). Other precision pv studies of the weak interactions of electrons and nuclei have been carried out with atoms. Despite their being at lower momenta, where the pv effects are smaller, ∼10−11, these experiments have reached the incredible precision of 1/2%, which atomic theory has yet to equal; theoretical errors are at the level of ∼1% [15]. At this level of precision the atomic experiments provide meaningful tests of the standard model. The dominant weak interaction term is aµV µ where the lower case aµ is the axial current of the electron and V µ is the vector current of the nucleus. This vector current is coherent over the nucleus, giving an effective charge QW = (1 −4 sin2 θW)Z −N (10) which is large for heavy atoms. The measurement on Cs, a one valence electron atom, at the 1/2% level, give QW = −72.35 ±0.27exp ±0.54th. [16] This results in an s-parameter [17] s = −1.0 ± (0.3)exp ± (1.0)th , where s is one of the parameters which measures deviation from the standard model. For instance, this gives a lower limit on the mass of a second Z boson of ∼500 GeV. [17] It has been proposed that pv measurements on a series of isotopes would allow one to obtain neutron radii for these nuclei. This is because QW is primarily sensitive to the neutron distribution [see Eq.(10)] and sin2 θW is well known from other measurements. This method may prove to be the most accurate means of measuring differences of neutron and proton radii of nuclei. [18] The term vµAµ is much smaller than aµV µ because for the electron vµ ∝(1−4 sin2 θW) ∼ 0.1 and only a single nucleon contributes to Aµ ∝< ⃗σ >, the nuclear spin. Thus, the asym- metry is reduced by ≥500. The atomic measurements of this term make use of the hyperfine structure, which is due to the nuclear spin. This term has not yet been detected because it is hidden by the stronger nuclear anapole moment. What is this? It is a parity violating moment discovered by Zel’dovich in 1957. [19] (I discovered it independently in 1973, [20] but did not name it, but simply called it", + "nuclear spin. This term has not yet been detected because it is hidden by the stronger nuclear anapole moment. What is this? It is a parity violating moment discovered by Zel’dovich in 1957. [19] (I discovered it independently in 1973, [20] but did not name it, but simply called it an axial coupling of the photon.) The anapole exists only for virtual photons [20] that penetrate the nucleus. It is, in reality, a combination of an electromagnetic interaction and a pv component of the nuclear wavefunction. This combination gives rise to a current similar to that in a winding on the surface of a doughnut; see Fig.3. [21] 6 Fig.3. Surface current ( ⃗J) on a doughnut, producing a toroidal magnetic field ( ⃗B) and anapole(⃗T). The most general form for an axial coupling of the photon to a nucleon or other spin 1/2 particle is Aµ = ¯ψ̸ qqµ −q2γµ M2 γ5ψa, (11) where a is the magnitude of the anapole moment. Note the need for the q2 dependence to preserve gauge invariance. The first term vanishes for a conserved vector current of the electron. Combined with the 1/q2 propagator, the second term is a contact one, unlike the electromagnetic interaction. The anapole moment can be written as ⃗a = −π Z d3rr2⃗j(r) = 1 e GF √ 2(−1)J+1/2+ℓ J + 1 2 J(J + 1) ⃗Jκa, (12) where J is the spin of the nucleus and κa is a dimensionless constant which is a measure of the anapole moment. The r2 allows ⃗a to be produced by surface currents; it comes from the q2 dependence of the axial coupling. The anapole is a purely nuclear term which is measured in atoms. The effect is of order αGF, but it is proportional to the square of the nuclear radius and therefore A2/3. In Cs and heavier atoms, it is larger by almost an order of magnitude than the asymmetry due to neutral weak currents. The recent measurement in atomic Cs at the 1/2% level [16] has discovered the anapole moment in the 6S →7S transition; the experimenters find it through a difference in the hyperfine F = 4 →3 and F = 3 →4 transitions of 0.077 ± .011 mV/cm. This gives κa = 0.364±.062. The theoretical value is sensitive to the pv π - nucleon coupling and gives [22] fπ = 9.5 ± 2.1exp ± 3.5th × 10−7. (13) By comparison, nucleon pv experiments in 18F are sensitive only to fπ and give [23] fπ ≤1.5 × 10−7. (14) There appears to be a clear discrepancy between these two determinations of fπ. However, the value determined from the anapole moment depends alo on the pv couplings of other mesons, primarily the ρ, to the nucleon via the combination ∼fπ + 1 2fρ. It could be that the ρ-N pv constant, determined from other pv experiments in light nuclei and the N-N system and consistent with each other and theory, is not correct; or perhaps some other effect has been omitted in the anapole calculations. The so-called ”best value” of fπ,", + "1 2fρ. It could be that the ρ-N pv constant, determined from other pv experiments in light nuclei and the N-N system and consistent with each other and theory, is not correct; or perhaps some other effect has been omitted in the anapole calculations. The so-called ”best value” of fπ, based on a quark model calculation is [23] fπ ≃(4.6 ± 4.6) × 10−7, where the error is often neglected. Nevertheless, there appears to be an a (experimental?) discrepancy which remains to be resolved. 7 In addition to all of these studies, the electron has been used in studies of time reversal invariance (TRI). Specifically, searches for an atomic electric dipole moment in 199Hg are sensitive to any simultaneous pv and TRI in this system. E.N. Fortson et al. [24] obtain dE(199Hg) = −(1 ± 2.4 ± 3.6) × 10−28e −cm , (15) or | dE(199Hg) |≤8.7 × 10−28e −cm (95% confidence level). (16) This experiment has about the same sensitivity as measurements of the neutron electric dipole moment and gives θ ≤10−10, where θ is defined by the term L = −θ α∫ ∀πǫµναβGµνGαβ, (17) which would be the strong CP problem, except that θ ≪1. But no one knows why θ should be so small. At low energies, the pion is the embodiment of QCD and the electric dipole moment comes about from a T-odd π - N coupling [25,26] L = −{T π ⪯ ψ⃗τψ · ⃗φ , and f T π ≤1011 from the upper limits of both the neutron and 199Hg electric dipole moments. I hope that I have convinced you that, despite its age, the electron and its interaction with nucleons and nuclei remains a very useful tool for testing fundamental interactions and learning important information about the structure of hadrons. 8 REFERENCES [1] See e.g., I.S. Towner and J.C. Hardy in Symmetries and Fundamental Interactions in Nuclei, ed. W.C. Haxton and E.M. Henley (World Sci., Singapore) 1995, p. 183. [2] T Minamisono (pvt. communic.). [3] S.J. Freedman (pvt. communic.) [4] D.H. Wilkinson, Nucl. Phys. A511 ,301 (1990) and Z. Phys. A348 ,1291 (1994). [5] W.C. Haxton (pvt. communic.) [6] F. Boehm, ibid ref.1, p.67. [7] L. de Braekeleer (pvt. communic.) [8] M. Allet et al., Phys. Rev. Lett. 68, 572 (1992) and J. Sromicki et al., Phys. Rev.C53 932 (1995). [9] S.P. Rosen, ibid ref. 1, p251, and references therein. [10] M. Moe and S.P. Vogel, Ann. Rev. Nucl. Part. Sci. 44, 247 (1994). [11] E.M. Henley, Revista Mexicana Fis. 40, 31 (1994) and R. McKeown and M.J. Musolf, ibid ref. 1, p.337. [12] P.A. Souder et al., Nucl. Phys. A527, 695c (1991). [13] B. Mueller et al., Phys. Rev. Lett. 78, 3824 (1997). [14] M.J. Musolf et al., Phys. Rep. 239, 1 (1994). [15] S.A. Blundell, J. Sapirstein, and W.R. Johnson, Phys. Rev. D45, 1602 (1992), V.A. Dzuba et al., J. Phys. (London) B20, 3297 (1987); P.A. Frantsuzo and I.B. Khriplovich Z. Phys., D7, 297 (1988). [16] C.S. Wood et al., Science 275, 1759 (1997). [17] J.L. Rosner, Phys. Rev. D53, 2724 (1996); V.A.", + "S.A. Blundell, J. Sapirstein, and W.R. Johnson, Phys. Rev. D45, 1602 (1992), V.A. Dzuba et al., J. Phys. (London) B20, 3297 (1987); P.A. Frantsuzo and I.B. Khriplovich Z. Phys., D7, 297 (1988). [16] C.S. Wood et al., Science 275, 1759 (1997). [17] J.L. Rosner, Phys. Rev. D53, 2724 (1996); V.A. Dzuba, V.V. Flambaum, and O.P. Sushkov, LANL Archives hep-ph 19709251, Sept. 1997 and Phys. Rev. A (to be pub- lished). [18] S. Pollock, E.N. Fortson, and L. Wilets, Phys. Rev. C46, 2587 (1992). [19] Ia. B. Zel’dovich, Zh. Eksp.Teor.Fiz. 33, 1531 (1957), tr. Sov. Phys. JETP 6, 1184 (1957). [20] E.M. Henley, A.H. Huffman, and D.U.L. Yu, Phys. Rev. D7, 943 (1973). [21] R.R. Lewis, Comm. At. Mol. Phys. 31, 67 (1995). [22] V.V. Flambaum and D.W. Murray, Phys. Rev. C56, 1641 (1997); V.K. Dimitriev and V.B. Telitsin, Nucl. Phys. A613, 237 (1997). [23] E.G. Adelberger and W.C. Haxton, Ann. Rev. Nucl. Part. Sci. 35, 501 (1985); B. Desplanques in Parity and Time Reversal Compound Nuclear States and Related Topics, ed. N. Auerbach and J.D. Bowman (World Sci., Singapore) 1996, p.98. [24] J.P. Jacobs et. al., Phys. Rev. Lett. 71, 3782 (1993) and Phys. Rev. A52, 3521 (1995). [25] R.J. Crewther et al., Phys.Lett B88, 123 (1979), B91, 487(E) (1980). [26] P. Herczeg, ibid ref. 1, p. 89. 9 This figure \"fig1-1.png\" is available in \"png\" format from: http://arxiv.org/ps/nucl-th/9801021v1 This figure \"fig1-2.png\" is available in \"png\" format from: http://arxiv.org/ps/nucl-th/9801021v1 This figure \"fig1-3.png\" is available in \"png\" format from: http://arxiv.org/ps/nucl-th/9801021v1", + "Resonating valence bond states in an electron-phonon system Zhaoyu Han and Steven A. Kivelson Department of Physics, Stanford University, Stanford, California 94305, USA We study a simple electron-phonon model on square and triangular versions of the Lieb-lattice using an asymptotically exact strong coupling analysis. At zero temperature and electron density n = 1 (one electron per unit cell), for various ranges of parameters in the model, we exploit a mapping to the quantum dimer model to establish the existence of a spin-liquid phase with Z2 topological order (on the triangular lattice) and a multi-critical line corresponding to a quantum critical spin liquid (on the square lattice). In the remaining part of the phase diagram, we find a host of charge-density-wave phases (valence-bond solids), a conventional s-wave superconducting phase, and with the addition of a small Hubbard U to tip the balance, a phonon-induced d-wave superconducting phase. Under a special condition, we find a hidden pseudo-spin SU(2) symmetry that implies an exact constraint on the superconducting order parameters. The electron-phonon interaction plays an essential role in the physics of quantum materials, e.g. for Bardeen- Cooper-Schrieffer superconductivity (SC) and typical charge or bond-density-wave ordering [1–3]. In the past few years, it has become increasingly clear that electron- phonon interactions can also induce a variety of more ex- otic behaviors and novel quantum phases [4–22], includ- ing those that are typically associated with strong repul- sive interactions, e.g. anti-ferromagnetism (AF) [10–13] and d-wave SC [14–22]. In this letter, we study a simple electron-phonon model, the “Holstein-Lieb” model (illustrated in Fig. 1), for which it is possible to obtain well-controlled results concerning the ground-state phase diagram (summarized in Fig. 2) through the use of an asymptotic strong- coupling expansion. Certain of the phases are interesting but not surprising – for instance, phonon-stabilized bipo- larons can order (localize) to form a variety of valence bond solid (VBS) phases, or when the strongly coupled sites lie above the Fermi energy, they act as “negative U centers” that mediate SC pairing [23]. More unexpect- edly, there is a range of parameters in which the problem maps onto a quantum dimer model [24, 25] introduced by Rokshar and Kivelson (RK), and thus exhibits a variety of exotic “resonating valence-bond” (RVB) phases known FIG. 1. An illustration of the Lieb lattices studied in this paper, and the “Holstein-Lieb” model in Eq. 1. to arise there, including (on the triangular lattice) a Z2 topologically ordered phase and (on the square lattice) a multicritical point which acts as the mother-state for an infinite hierarchy of incommensurate phases. With this concrete example, we hope to suggest new avenues for the search for materials supporting “spin liquid” phases [26] in systems with relatively strong electron-phonon cou- plings. To date, this effort has been almost entirely fo- cused on studies of frustrated anti-ferromagnets. We consider a Hamiltonian in which electrons can oc- cupy orbitals on the vertices of the lattice, j, or on the bond centers between pairs of nearest-neighbor sites, ⟨ij⟩: ˆH = −t X ⟨ij⟩σ h ˆf † ⟨ij⟩σ (ˆciσ + ˆcjσ) + h.c.", + "fo- cused on studies of frustrated anti-ferromagnets. We consider a Hamiltonian in which electrons can oc- cupy orbitals on the vertices of the lattice, j, or on the bond centers between pairs of nearest-neighbor sites, ⟨ij⟩: ˆH = −t X ⟨ij⟩σ h ˆf † ⟨ij⟩σ (ˆciσ + ˆcjσ) + h.c. i + X ⟨ij⟩ \u0010 E + α ˆX⟨ij⟩ \u0011 ˆn⟨ij⟩+ ˆHph (1) where ˆciσ ( ˆf⟨ij⟩σ) annihilates a spin-σ electron on the or- bital at site i (bond ⟨ij⟩), and ˆni or ˆn⟨ij⟩are the electron numbers on the corresponding orbitals. ˆX⟨ij⟩are the co- ordinate operators of optical phonons on bonds described by the Hamiltonian: ˆHph = X ⟨ij⟩ \" K ˆX2 ⟨ij⟩ 2 + ˆP 2 ⟨ij⟩ 2M # . (2) By rescaling variables it is easy to see that there are precisely four independent energy scales in the problem: the band-structure scale t, the charge transfer gap E, the phonon frequency, ω0 ≡ p K/M, and the phonon- induced electronic attraction, Ue-ph ≡α2/K, which is the relevant measure of the electron-phonon coupling strength. We will perform a controllable strong-coupling analysis assuming |t| to be a small energy scale (the pre- cise meaning of this assumption will be made clear be- low), and obtain effective theories for the active degrees of freedom in different parameter regimes. At the end of the paper, we will discuss the robustness of the results in the presence of additional couplings that are likely to be present in candidate materials. arXiv:2210.16321v2 [cond-mat.str-el] 3 May 2023 2 Methods. The effects of strong electron-phonon cou- pling can best be addressed following a unitary transfor- mation [27] ˆU ≡exp  i α K X ⟨ij⟩ ˆP⟨ij⟩ˆn⟨ij⟩  , (3) that transforms the Hamiltonian into: ˆU † ˆH ˆU = −t X ⟨ij⟩σ h ˆD⟨ij⟩ˆf † ⟨ij⟩σ (ˆciσ + ˆcjσ) + h.c. i + X ⟨ij⟩ \u0014 Eˆn⟨ij⟩−Ue-ph 2 ˆn2 ⟨ij⟩ \u0015 + ˆHph (4) The resulting theory contains a unitary operator ˆD⟨ij⟩≡ e−i ˆ P⟨ij⟩α/K that displaces the phonon coordinate by α/K. We will perform a perturbative analysis treating the sec- ond line in Eq. 20 as the unperturbed Hamiltonian, ˆH0, and the first line as perturbations, ˆH′. Under most cir- cumstances, ˆH0 has an extensive ground-state degener- acy, so we use degenerate perturbation theory to derive an effective model acting in the degenerate subspace. We note that all the virtual processes, including those with phonon excitations, are included in this analysis. We keep terms in powers of t to the lowest order needed to resolve the degeneracy, and the validity of each model will be analyzed in each scenario. Since ˆH can be de- fined on any lattice in any dimension, so can the result- ing effective theories. Here, to be explicit, we will confine ourselves to the square and triangular lattices in two di- mensions. The first step in our analysis is to identify the degen- erate ground-state manifold of ˆH0, which we call H0; we will restrict our attention to the range of electron den- sities per unit cell, 0 < n ≤2. For convenience, we", + "square and triangular lattices in two di- mensions. The first step in our analysis is to identify the degen- erate ground-state manifold of ˆH0, which we call H0; we will restrict our attention to the range of electron den- sities per unit cell, 0 < n ≤2. For convenience, we define E1 ≡E −Ue-ph/2 and E2 ≡2E −2Ue-ph to rep- resent the energies of a singly or doubly occupied bond orbital, and E12 ≡E1 −E2 = 3Ue-ph/2 −E to represent their energy difference. Because E2 −2E1 = −Ue-ph is al- ways negative, singly occupied bond orbitals are always disfavored. Therefore, depending on the sign of E2, bond or site orbitals are favored, so that all possible occupa- tion configurations of bond dimers or site electrons form a basis of H0. For the case where bond dimers are active degrees of freedom (E2 ≲0), there are two sorts of terms that will be generated by the perturbative analysis: There are diago- nal terms (dimer potential energy) and off-diagonal terms (dimer kinetic energy). Since the phonon displacements are different for different dimer configurations, all off- diagonal terms must vanish in the limit of non-dynamical phonons, ω0 = 0. Specifically, the amplitude of any pro- cess in which a dimer relocates onto or offof a bond is accompanied by a Frank-Condon factor F [28] defined as F ≡⟨0| ˆD2|0⟩= e−X, (5) FIG. 2. Schematic T = 0 phase diagram for n = 1 in the small |t| limit as a function of the dimensionless ratios of pa- rameters in the model, Eq. 1; the vertical axis quantifies the degree of retardation and the horizontal axis the strength of the electron-phonon coupling. The blue region on the left top is described by the quantum dimer model in Eq. 7. The yel- low region on the right is described by the weakly interacting theory in Eq. 27. The red region in the middle is described by Eq. 9 and is confined to a narrow window. The grey region on the left bottom is described by the effective bosonic Hamil- tonian, Eq. 6, where controlled analysis of the ground-state phases is missing. The RK line, which corresponds to the ex- actly solvable point V2 = τ2 in Eq. 7, occurs with moderately large retardation, Ue-ph/ω0 = O \u0000ln E12 t \u0001 , as long as E/Ue-ph is not close to 1. The Z2 symbol represents a Z2 spin liquid phase on the triangular lattice, and IC stands for possible incommensurate crystalline phases on the square lattice. where |0⟩is the ground state of the phonon Hamiltonian on the bond, and X ≡Ue-ph/ω0 is a dimensionless factor quantifying the degree of retardation, and the displace- ment operator ˆD is squared since the occupancy of the orbital changes by two electrons. This factor becomes arbitrarily small in the limit of strong retardation. On the other hand, the potential terms always only receive O(1) factors from the virtual phonon fluctuations. Below we derive the effective theories and obtain ex- pressions as functions of the bare energy scales for the coupling constants that", + "This factor becomes arbitrarily small in the limit of strong retardation. On the other hand, the potential terms always only receive O(1) factors from the virtual phonon fluctuations. Below we derive the effective theories and obtain ex- pressions as functions of the bare energy scales for the coupling constants that arise in low-order perturbation theory in t. The effective Hamiltonians are given in Eqs. 6, 27 & 9. The asymptotic expressions and the limiting behaviors in the small and large ω0 limits of the effective couplings are given in Table. II. Their explicit expressions and derivations are deferred to Supplemental Materials [29]. Ue-ph > E: dimer models. In this case, the energy necessary for breaking a dimer is E12; thus the expan- sion series in t is controllable as long as t ≪E12. For a dimer on bond ⟨ij⟩, we define the annihilation operator as ˆb⟨ij⟩≡ˆf⟨ij⟩↑ˆf⟨ij⟩↓and the dimer occupation number operator ˆnb ⟨ij⟩≡ˆb† ⟨ij⟩ˆb⟨ij⟩= 0, 1. To the fourth order in 3 anti-adiabatic adiabatic ω0 →∞, X →0 ω0 →0 , X →∞ teff∼t2 E1 t2 E1 t2 E τ0 ∼2t2F E1 2t2 E1 √ πXt2 E1 e−X →0 τ1 ∼t4F 2 E3 12 4t4(2Ue-ph−E) E2 12|E2|Ue-ph 4 √ 2πXt4 E2 2 Ue-ph e−2X →0 V1 ∼ t4 E3 12 4t4(5Ue-ph/2−E) E3 12Ue-ph 2t4(4Ue-ph−E) (2Ue-ph−E)3Ue-ph Jeff∼t4 E3 1 4t4Ue-ph E3 1 E2 t4U3 e-ph 2E2 1 E4 TABLE I. Asymptotic expressions and limiting behaviors of the coefficients in the effective theories in Eqs. 6, 27 & 9. Each limiting behavior is evaluated in the region of validity of the corresponding effective Hamiltonian. t, we obtain the following model for the dimers: ˆHb = X ⟨ijk⟩ h −τ1 \u0010 ˆb† ⟨ij⟩ˆb⟨jk⟩+ h.c. \u0011 + V1ˆnb ⟨ij⟩ˆnb ⟨jk⟩ i (6) where the summation is over all pairs of nearest-neighbor bonds with a single vertex in common, and it is implicit that we have omitted terms of order t6 and higher, to which we shall return shortly. Therefore, we obtain a hard-core boson model on the bond lattice with repulsive interactions. As shown in Table. II, when X ≲1 it follows that τ1 ∼V1, so this is an interacting problem with no small parameter to give theoretical control. We label this re- gion “Interacting dimers” in Fig. 2. For n ≪1, the ground state is presumably an s-wave superfluid, inde- pendent of the details of the lattice structure and the in- teractions. As n approaches 1, the balance between the interactions and the kinetic terms becomes more subtle. Similar models have been studied on several lattices [30– 35], where various superfluid, charge and supersolid or- ders were found. We expect analogous phases to arise in the present model. However, when X is large, such that F ≪1, the V1 term is dominant over τ1. Those ground states of V1 within H0 form an emergent low-energy (still extensively degenerate) subspace, H1, in which, as in the RK quan- tum dimer models, no more than one dimer can touch a given site. Within H1, we further perform perturbative analysis on square and triangular lattices and obtain the", + "ground states of V1 within H0 form an emergent low-energy (still extensively degenerate) subspace, H1, in which, as in the RK quan- tum dimer models, no more than one dimer can touch a given site. Within H1, we further perform perturbative analysis on square and triangular lattices and obtain the RK model as the effective model: ˆHRK =V2 X \u0002 + \u0003 −τ2 X \u0002 + \u0003 (7) where the ket (bra) represents a pair of annihilation (cre- ation) operators on the thickened bonds, and the summa- tion is over all possible four-sided plaquettes. To leading order in τ1/V1, it is easy to see that τ2 = 4τ 2 1 V1 ∼ t4 E3 12 F 4. In terms of the same expansion, one would conclude that V2 = O \u0000τ 5 1 /V 4 1 \u0001 is always small compared to τ2. How- ever, since we are simultaneously assuming both t and F are small, we need to consider terms (not shown in Eq. 6) that are higher order in t, but which are not suppressed by F (potential terms). (See Fig. 3 for illustrations of the virtual processes that contribute to the effective theory.) Since it involves such high-order processes, it is not worth writing out explicitly the results. What is essential is that the leading term only contributes to V2 and it is eighth order in t and positive, i.e. V2 ∼ t8 E7 12 [36]. From the perturbative perspective, there are two types of leading virtual process, both of which contribute pos- itively to V2: If two dimers occupy two parallel sides of a four-sided plaquette, the lowest order virtual pro- cess that non-trivially connects them is a ring exchange of electrons, which results in a fermion minus sign that turns what would have been an energy gain into a cost. On the other hand, if only one dimer occupies a side of a plaquette, there is an allowed virtual process in which a single electron travels around the plaquette - this energy- gaining process is blocked when there are two dimers on the plaquette. Both processes are eighth order in t on the lattices we are considering. [We have performed ex- act diagonalization studies on small clusters in the limit ω0 = 0 as a check of these conclusions.] Formally, we can consider ˆHRK to be the effective Hamiltonian in an asymptotic limit where F ∼|t|/E12 ∼ δ ≪1, such that the couplings τ2 and V2 are both of order δ8, with relative magnitudes that can be tuned in a wide range - for instance by varying ω0. All omitted terms are higher order in powers of δ. The zero-temperature phase diagrams of the quantum dimer models Eq. 7 on the square and triangular lat- tices have been solved, with results we briefly summarize here. For both cases, the line along which τ2 = V2 is special (corresponding to the RK point [24, 25]) and is labeled “RK” in Fig. 2. Here, an exact ground state is an equal amplitude superposition of all dimer configu- rations in", + "been solved, with results we briefly summarize here. For both cases, the line along which τ2 = V2 is special (corresponding to the RK point [24, 25]) and is labeled “RK” in Fig. 2. Here, an exact ground state is an equal amplitude superposition of all dimer configu- rations in a given topological sector, i.e. a short-ranged RVB state [26]. The ground states in different topological sectors are exactly degenerate, which leads to topological FIG. 3. Illustration of the virtual processes (arrows indicate the direction of electron hops) on a four-sided plaquette con- tributing to the terms in the RK effective theory in Eq. 7. The first class contributes to τ2 while the latter two contribute to V2. 4 degeneracy on compact manifolds. Through exact evalu- ations of the dimer correlations [37–39], it is known that this point is gapless on the square lattice and gapped on the triangular lattice. Further numerical and analytical investigations have fleshed out the full phase diagram of this model. On the square lattice, for the model defined in Eq. 7, the RK point is a critical point separating two different VBS states: staggered (for V2 > τ2 > 0) and plaquette (for 0 < V2 < τ2) [40, 41]. More generally, it is an unstable multi-critical point described by the quantum Lifshitz model [39, 42], near which a wide class of perturbations can induce incommensurate crystalline phasesthrough a mechanism known as Cantor deconfinement [42, 43]. We label the region that may host such additional phases “IC” in Fig. 2. [44] On the triangular lattice, the RK point lies on the boundary of a phase that exhibits Z2 topological order in a range of νcτ2 < V2 < τ2 with νc ≲ 0.8 (marked Z2 in Fig. 2); two different VBS occur for other ranges of parameters: staggered (for V2 > τ2 > 0) and √ 12 × √ 12 (for 0 < V2 < νcτ2) [45, 46], which we also refer to as “plaquette VBS” in the schematic phase diagram Fig. 2. The Z2 spin liquid is known to have several types of excitations: spinons, holons, and visions. In the current case, while visions have relatively low creation energy ∼τ2, the creation of spinons or holons carries a large energy cost ∼E2 or ∼Ue-ph in order to break a dimer. Therefore, when lightly doping holes into the system near the RK point such that |n −1| = x ≪1, we will likely have dimer vacancies as charge carriers leading to con- densation with SC Tc determined by the coherence scale, Tc ∼xτ1 ∼x t6 E5 12 [47]. For the square lattice RK model, exactly at the RK point, there are also gapless “resonon” excitations with momenta near (π, π) and a quadratic dis- persion. Since the motion of the dimers is tied to that of the phonons, the emergence of such excitations should be observable in measurements of the phonon spectrum, e.g. through neutron scattering. Ue-ph < E: weakly interacting model. In this case, the effective model is expressed in terms of site elec- trons. To", + "motion of the dimers is tied to that of the phonons, the emergence of such excitations should be observable in measurements of the phonon spectrum, e.g. through neutron scattering. Ue-ph < E: weakly interacting model. In this case, the effective model is expressed in terms of site elec- trons. To fourth order in t, the effective Hamiltonian is ˆHc = −teff X ⟨ij⟩σ \u0010 ˆc† iσˆcjσ + h.c. \u0011 −2Jeff X ⟨ij⟩ ˆn[i+j]↑ˆn[i+j]↓ (8) where ˆn[i+j]σ ≡ \u0010 ˆc† iσ + ˆc† jσ \u0011 (ˆciσ + ˆcjσ) /2 is the number of electrons in a bonding orbital between sites i and j. Since any virtual movement of electrons necessarily costs E1 in the intermediate state, the expansion is valid as long as |t| ≪E1. As can be seen from Table II, teff is second order in t and Jeffis fourth order. Thus, we should consider this theory in its weak coupling limit, Jeff≪teff. With detailed discussion in Supplemental Materi- als [29], we analyze the weak-coupling instabilities in the context of a Hartree-Fock mean-field analysis that is rea- sonable in this limit. We find that, on square and triangu- lar lattices, s-wave SC is always the dominant instability for n < 2. However, when n ≈1 on the square lattice, there is also a d-wave pairing state that is only moder- ately subdominant to the dominant s-wave channel. This competition between the s- and d-wave paired states can be tuned by the additional weak Hubbard repulsion on site orbitals, Uc; when Uc ≳2.2Jeff, the d-wave paired state has the lower variational energy. Exactly at n = 1, there is also an AF instability, which is also subdominant to the s-wave SC instability, but which is favored over all superconducting states when Uc > 2Jeff. Ue-ph ≈E: monomer-dimer model. In the narrow region |E2| ≲τ0, where τ0 (again given in Table II) is the matrix element for converting a pair of site electrons to a pair of bond electrons, both c and f orbitals are active. The effective Hamiltonian in this case is ˆHbc = X ⟨ij⟩ h teff \u0010 2ˆnb ⟨ij⟩−1 \u0011 2ˆn[i+j] + (E2 −4teff) ˆnb ⟨ij⟩ i + τ0 X ⟨ij⟩ h ˆb† ⟨ij⟩(ˆci↑+ ˆcj↑) (ˆci↓+ ˆcj↓) + h.c. i (9) In this regime, no controllable analysis can be performed. A mean-field analysis, treating b and c as decoupled, sug- gests an s-wave SC phase - presumably one that connects to the corresponding phase in the Ue-ph < E case. It is interesting to note that, for arbitrary t, there is a hidden pseudo-spin SU(2) symmetry in the original problem when E2 = 0 and X →0 (marked with the red circle in Fig. 2), which is a generalization of that of the Hubbard model on bipartite lattices [48, 49]. This symmetry implies, in the thermodynamic limit, for the ground state of the system at any filling: 1 N 2 site * X i ˆci↑ˆci↓− X ⟨ij⟩ ˆf⟨ij⟩↑ˆf⟨ij⟩↓ 2+ = 0 (10) which is, as discussed in detail in Supplemental Materi- als [29], difficult to satisfy in any SC state that", + "symmetry implies, in the thermodynamic limit, for the ground state of the system at any filling: 1 N 2 site * X i ˆci↑ˆci↓− X ⟨ij⟩ ˆf⟨ij⟩↑ˆf⟨ij⟩↓ 2+ = 0 (10) which is, as discussed in detail in Supplemental Materi- als [29], difficult to satisfy in any SC state that does not have space-dependent oscillations in sign. Outlook. The derivations of the effective theories can be easily generalized to include strong (in compari- son to t) or even infinite Hubbard repulsion, Uc, on the site orbitals. In that case, the model corresponding to Ue-ph < E at n ≲1 is a t-J model (with no double oc- cupancy constraint on site orbitals). The AF coupling J in this model is enhanced by the effective attraction on the bond orbitals, and an extra nearest-neighbor density- density repulsion interaction is induced by phonon vir- tual fluctuations. On the other hand, the dimer models for Ue-ph > E are not qualitatively changed by the pres- ence of a repulsive Hubbard interaction on site and bond 5 orbitals, nor weak further-ranged hopping and electron- phonon coupling, as long as H0 is unaffected. In considering the search for “spin liquid” phases in real materials featuring significant electron-phonon cou- plings, we summarize the key ingredients that we think are crucial for the mechanism revealed in this work: 1. atomic-scale structures with electronically active atoms on both vertices and the bridging sites between them (it is encouraging to note that a large class of real ma- terials have this feature [50–52]); 2. strong coupling to phonon modes on bonds that allow the formation of bipo- larons localized on bonds; 3. a moderately large degree of retardation that suppresses the quantum hopping rel- ative to interactions and thus leads to constraints on the low-energy Hilbert space. 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Lee, Phys. Rev. B 107, L041103", + "231 (2020). [56] H. Yao and S. A. Kivelson, Phys. Rev. Lett. 99, 247203 (2007). [57] K.-S. Kim, Phys. Rev. B 107, L140401 (2023). [58] J. Sous and M. Pretko, Phys. Rev. B 102, 214437 (2020). [59] Z.-X. Li, S. G. Louie, and D.-H. Lee, Phys. Rev. B 107, L041103 (2023). RESONATING VALENCE BOND STATES IN AN ELECTRON-PHONON SYSTEM: SUPPLEMENTARY MATERIALS The effective coefficients For the second-order processes, it suffices to compute the following factor: g±(ξ) ≡e(1∓1)X/2 X n ⟨0|D( p X/2)|n⟩⟨n|D(∓ p X/2)|0⟩ 1 + n/ξ1 =ξ Z ∞ 0 dt exp \u0014 −ξt ± X 2 \u0000e−t −1 \u0001\u0015 (11) where D( p X/2) is the displacement operator written in the coherent state basis. To obtain the second equality we have used the Feynman’s trick 1/u = R ∞ 0 dte−ut. Similar derivations can be found in the Supplemental Materials of anti-adiabatic ω0 →∞ adiabatic ω0 →0 teff= t2 E1 g+(Y ) t2 E−Ue-ph/2 t2 E τ0 = 2t2F E1 g−(Y ) 2t2 E−Ue-ph/2 √πt2 E−Ue-ph/2 q Ue-ph ω0 e−Ue-ph/ω0 →0 τ1 = 2t4F 2 E2 12 h 2 |E2|g2 −(W) + 1 Ue-ph g3−(W, X) i 2t4(2Ue-ph−E) (3Ue-ph/2−E)2(Ue-ph−E)Ue-ph 2t4 (Ue-ph−E)2 q π 2Ue-phω0 e−2Ue-ph/ω0 →0 V1 = 4t4 E2 12 h 1 E12 g1(W) + 1 Ue-ph g2(W, X) i 4t4(5Ue-ph/2−E) (3Ue-ph/2−E)3Ue-ph 2t4(4Ue-ph−E) (2Ue-ph−E)3Ue-ph Jeff= 4t4 E2 1 h 2 E2 G+(Y, Z) − 1 E1 g1(Y ) + 1 E1 G′ −(Y ) i 2t4Ue-ph (E−Ue-ph/2)3(E−Ue-ph) t4U3 e-ph 2(E−Ue-ph/2)2E4 g±(ξ) ≡ξ R 1 0 dz z zξ exp \u0002 ± X 2 (z −1) \u0003 g1(ξ) ≡ξ3 R 1 0 Q3 i=1 dzi zi (z1z2z3)ξ exp \u0002 X 2 (z1z2 + z3 −2) \u0003 g2(ξ1, ξ2) ≡ξ2 1ξ2 R 1 0 Q3 i=1 dzi zi (z1z3)ξ1zξ2 2 exp \u0002 X 2 (z1z2z3 + z2 −2) \u0003 g3±(ξ1, ξ2) ≡ξ2 1ξ2 R 1 0 Q3 i=1 dzi zi (z1z3)ξ1zξ2 2 exp \u0002 ± X 2 (z1z2 + z2z3 −2) \u0003 G±(ξ1, ξ2) ≡ξ2 1ξ2 R 1 0 Q3 i=1 dzi zi (z1z3)ξ1zξ2 2 exp X 2 [∓(z1 + z3) + z2 (1 ± z1) (1 ± z3) −2] G′ −(ξ) ≡ξ3 R 1 0 Q3 i=1 dzi zi (z1z3)ξ exp \u0002 X 2 (z1 + z3 −2) \u0003 \b exp \u0002 X 2 z2 (1 −z1) (1 −z3) \u0003 −1 TABLE II. Expressions and limiting behaviors of the coefficients in the effective theories in Eqs. 7, 9 & 11 in terms of the dimensionless parameters X ≡Ue-ph/ω0, Y ≡E1/ω0, Z ≡|E2|/ω0, and W ≡E12/ω0. Each limiting behavior is evaluated in the region of validity of the corresponding effective Hamiltonian. 7 Ref. [5]. Using these factors one can express teff= t2 E1 X n ⟨0|D( p X/2)|n⟩⟨n|D(∓ p X/2)|0⟩ 1 + ω0n/E1 = t2 E1 g+(E1/ω0) (12) and similarly τ0 = 2t2 E1 g−(E1/ω0)e−X. In anti-adiabatic limit ω →∞, X, ξ →0, the e−t term on the exponent of the integrand decays rapidly in the t ∼1/ξ scale, so we can neglect it and obtain G →e−X/2 ∼1. In adiabatic limit ω →0, X, ξ →∞, the e−t term on", + "= 2t2 E1 g−(E1/ω0)e−X. In anti-adiabatic limit ω →∞, X, ξ →0, the e−t term on the exponent of the integrand decays rapidly in the t ∼1/ξ scale, so we can neglect it and obtain G →e−X/2 ∼1. In adiabatic limit ω →0, X, ξ →∞, the e−t term on the exponent of the integrand decays slowly in the t ∼1/ξ scale, so we can expand it to e−t ≈1 −t and obtain G → 1 1±X/(2ξ). [For g−, the above adiabatic result only applies when X < 2ξ. When X ≈2ξ, g−→ p πX/4 in this limit. If X > 2ξ, g−→ξ p 4π/Xe+X/4 will become an exponentially large factor that will give a correction to the Frank-Condon Factor F(X) = e−X. For the cases studied in this paper, only X ≈2ξ case is physically relevant. ] For the fourth-order processes, it will be useful to evaluate the following quantity: I(ϵ1, . . . , ϵ4; a1, . . . , a4) ≡tr  X {ni} 4 O i=1 \u0000D(ai)|ni⟩e−ϵi⟨ni| \u0001   = Z Q4 i=1 d(αi, α⋆ i )d(βi, β⋆ i ) π8 exp −1 2 \" 4 X i=1 |αi|2 + 2|βi|2 + |αi + ai|2 −2α⋆ i βie−ϵi −a⋆ i α⋆ i + aiαi −2β⋆ i (αi + ai) # (13) where |n⟩is the n-th eigen state of the phonon Hamiltonian and D(a) = exp(aˆa† −a⋆ˆa) is the displacement operator. The second equality is obtained by inserting coherent state identities. Since we are interested in zero-temperature theory, we may set ϵ4 →∞. This leads to: I(ϵ1, ϵ2, ϵ3, ϵ4 = ∞; a1, . . . , a4) = exp − \" a1a2e−ϵ1 + a2a3e−ϵ2 + a3a4e−ϵ3 + a1a3e−ϵ1−ϵ2 + a2a4e−ϵ2−ϵ3 + a1a4e−ϵ1−ϵ2−ϵ3 + X i a2 i /2 # (14) Using this result, we now compute the factor associated with the fourth order processes that involve a sequence of displacements of a bond phonon described by η1η2η3η4, where ηi = ± corresponding to an electron jumping in or out of the bond site. Note that P4 i=1 ηi = 0 in order to go back to the ground state manifold. We then obtain Gη1η2η3η4(ξ1, ξ2) ≡ X {ni} ⟨0|D(η1 p X/2)|n1⟩⟨n1|D(η2 p X/2)|n2⟩⟨n2|D(η3 p X/2)|n3⟩⟨n3|D(η4 p X/2)|0⟩ (1 + n1/ξ1)(1 + n2/ξ2)(1 + n3/ξ1) =ξ2 1ξ2 Z ∞ 0 dt1dt2dt3 exp \u001a −ξ1(t1 + t3) −ξ2t2 −X 2 [η1η2 (e1 + e3) + η2η3 (e2 + e1e2e3) + η1η3 (e1e2 + e2e3) + 2] \u001b (15) where ei ≡e−ti. In anti-adiabatic limit ω →∞, X, ξi →0, G →1; In adiabatic limit ω →0, X, ξi →∞, G → 1 [1+X/(2ξ1)]2 1 1+(1+η1η2)X/ξ2 . For the virtual processes where the intermediate state only has only phonon excitation (but no electronic excitation), we will also need to evaluate: G′ η1,−η1,η3,−η3(ξ) ≡ X {ni} ⟨0|D(η1 p X/2)|n1⟩⟨n1|D(η2 p X/2)|n2⟩⟨n2|D(η3 p X/2)|n3⟩⟨n3|D(η4 p X/2)|0⟩ (1 + n1/ξ)n2/ξ(1 + n3/ξ) =ξ ∂Gη1,−η1,η3,−η3(ξ, ξ2) ∂ξ2 ξ2→0 (16) 8 In anti-adiabatic limit ω →∞, X, ξi →0, G →X 2 ; In adiabatic limit ω →0, X, ξi →∞, G →0 (no slower", + "G′ η1,−η1,η3,−η3(ξ) ≡ X {ni} ⟨0|D(η1 p X/2)|n1⟩⟨n1|D(η2 p X/2)|n2⟩⟨n2|D(η3 p X/2)|n3⟩⟨n3|D(η4 p X/2)|0⟩ (1 + n1/ξ)n2/ξ(1 + n3/ξ) =ξ ∂Gη1,−η1,η3,−η3(ξ, ξ2) ∂ξ2 ξ2→0 (16) 8 In anti-adiabatic limit ω →∞, X, ξi →0, G →X 2 ; In adiabatic limit ω →0, X, ξi →∞, G →0 (no slower than 1/X2). In the main text, we call G+ ≡G++−−= G−−++, and G−≡G+−+−= G−+−+. For the evaluation of the effective coeffecients for the dimer model, we need to compute g1(ξ) ≡ X {ni} ⟨0|D( p X/2)|n⟩⟨n|D(− p X/2)|0⟩⟨0|D( p X/2)|n′⟩⟨n′|D(− p X/2)|0⟩ (1 + n/ξ)2(1 + n′/ξ) =ξ3 Z ∞ 0 dt1dt2dt3 exp \u001a −ξ(t1 + t2 + t3) −X + X 2 [e1e2 + e3] \u001b (17) g2(ξ1, ξ2) ≡ X n,n′ ⟨0|D( p X/2)|n⟩⟨n|D(− p X/2)|0⟩⟨0|D( p X/2)|n′⟩⟨n′|D(− p X/2)|0⟩ (1 + n/ξ1)2(1 + n/ξ2 + n′/ξ2) =ξ2 1ξ2 Z ∞ 0 dt1dt2dt3 exp \u001a −ξ1(t1 + t3) −ξ2t2 −X + X 2 [e1e2e3 + e2] \u001b (18) g3±(ξ1, ξ2) ≡e(1∓1)X X n,n′ ⟨0|D( p X/2)|n⟩⟨n|D(∓ p X/2)|0⟩⟨0|D( p X/2)|n′⟩⟨n′|D(∓ p X/2)|0⟩ (1 + n/ξ1)(1 + n/ξ2 + n′/ξ2)(1 + n′/ξ1) =ξ2 1ξ2 Z ∞ 0 dt1dt2dt3 exp \u001a −ξ1(t1 + t3) −ξ2t2 ∓X ± X 2 [e1e2 + e2e3] \u001b (19) In anti-adiabatic limit ω →∞, X, ξ →0, gi →e∓X/2 ∼1; In adiabatic limit ω →0, X, ξ →∞, so g1 → 1 [1+X/(2ξ)]3 , g2 → 1 [1+X/(2ξ1)]2 1 1+X/ξ2 , and g3± → 1 [1±X/(2ξ1)]2 1 1±X/ξ2 . [For g3−, the above adiabatic result only applies when X < ξ2, 2ξ1. For the cases studied in this paper, ξ2 = X, ξ1 = W = ∆12/ω0 > X, and g3−→ √ πX/2 [1−X/(2W )]2 in the adiabatic limit. ] To keep the expressions short, we have made variable substitution zi ≡e−ti in Table. II. The pseudo-spin SU(2) symmetry In the anti-adiabatic limit X →0, the displacement operators ˆD play no role in the transformed Hamiltonian ˆHT , which will become simply a Hubbard model: ˆU † ˆH ˆU = −t X ⟨ij⟩σ h ˆf † ⟨ij⟩σ (ˆciσ + ˆcjσ) + h.c. i + Uc X i ˆni↑ˆni↓ + X ⟨ij⟩ \u0014 Eˆn⟨ij⟩−Ue-ph 2 ˆn2 ⟨ij⟩ \u0015 + Uf X ⟨ij⟩ ˆn⟨ij⟩↑ˆn⟨ij⟩↓ (20) where we have explicitly included other possible Hubbard interactions, Uc and Uf, in order to generalize the results here. Note that for arbitrary t, when 2E + Uf = 2Ue-ph + Uc, this Hamiltonian has an pseudo-spin SU(2) symmetry generated by the generators: ˆJz ≡ X i (ˆni −1) + X ⟨ij⟩ \u0000ˆn⟨ij⟩−1 \u0001 , ˆJ−≡ X i ˆci↑ˆci↓− X ⟨ij⟩ ˆf⟨ij⟩↑ˆf⟨ij⟩↓, ˆJ+ ≡ˆJ† − (21) These generators satisfy an SU(2) algebra, and they are eigen-operators [48] of the Hamiltonian in the sense that: h ˆHT , Jz i = 0 , h ˆHT , J± i = ±(−2µ + Uc)J± (22) where µ is the chemical potential that was left as implicit in the original expression of the Hamiltonian. Therefore, the Hamiltonian conserves both the z component and the total pseudo-spin: ˆJx ≡( ˆJ+ + ˆJ−)/2 , ˆJy ≡( ˆJ+ −ˆJ−)/(2i) , h", + ", J± i = ±(−2µ + Uc)J± (22) where µ is the chemical potential that was left as implicit in the original expression of the Hamiltonian. Therefore, the Hamiltonian conserves both the z component and the total pseudo-spin: ˆJx ≡( ˆJ+ + ˆJ−)/2 , ˆJy ≡( ˆJ+ −ˆJ−)/(2i) , h ˆHT , ˆJ2 x + ˆJ2 y + ˆJ2 z i = 0 (23) 9 This symmetry is a generalization of the pseudo-spin symmetry of the Hubbard model on bipartite lattices [49]. While we emphasize that the symmetry in the current model is present for arbitrary lattices (since f and c orbitals are always bipartite). When 2µ = Uc (corresponding to “half-filling”, one electron per orbital or n = 1 + z/2), this symmetry become a full SU(2) symmetry in the sense that all the three components of ˆJ are conserved, and it implies that the charge density operator ˆn and local pairing operator ˆc↑ˆc↓(or −ˆf↑ˆf↓) are symmetric. When away from “half-filling”, only Jz and J2 are conserved. In this case, we note that the Hamiltonian takes the form HSU(2) + BzJz where HSU(2) is the fully SU(2) symmetric part of the Hamiltonian and Bz = (−2µ + Uc) is the effective pseudo-Zeeman field. This implies that, all the energy levels can be written as En(J, Jz) = En(J) + JzBz. Therefore, for the ground state of the system, |Jz| = J must be satisfied in order to fully optimize the pseudo-Zeeman energy for any Bz. This means that, in the ground state: J(J + 1) = ⃗J2 = J2 z + J2 x + J2 y =⇒J2 x + J2 y = |Jz| (24) For the case we are considering, this becomes: X i ˆci↑ˆci↓− X ⟨ij⟩ ˆf⟨ij⟩↑ˆf⟨ij⟩↓ 2 = (1 + z/2 −n)Nsite (25) where n is the electron density per site, and z is the coordination number. Dividing both sides by Nsite and take thermodynamic limit Nsite →∞, we reach the conclusion: 1 N 2 site X i ˆci↑ˆci↓− X ⟨ij⟩ ˆf⟨ij⟩↑ˆf⟨ij⟩↓ 2 →0 (26) There are three possible physical conclusions from this result: 1. the ground state does not have off-diagonal long- range order (ODLRO); 2. it has uniform ODLRO, but ⟨2ˆci↑ˆci↓−P j ˆf⟨ij⟩↑ˆf⟨ij⟩↓⟩= 0 is exact for any i; 3. it has ODLRO, but the order parameter oscillate in space and average to zero, i.e. it is a pair density wave (PDW) state [55]. Therefore, this class of models may host PDW states, which remains to be explored in future studies. The Hartree-Fock mean-field analysis on the weakly interacting model In the case of Ue-ph < E, the effective model is expressed in terms of site electrons (c-electrons). To fourth order in t, the effective Hamiltonian is ˆHc = −teff X ⟨ij⟩σ \u0010 ˆc† iσˆcjσ + h.c. \u0011 −2Jeff X ⟨ij⟩ ˆn[i+j]↑ˆn[i+j]↓ (27) where ˆn[i+j]σ ≡ \u0010 ˆc† iσ + ˆc† jσ \u0011 (ˆciσ + ˆcjσ) /2 is the number of electrons in a bonding orbital between sites i and j. Since any virtual movement of electrons necessarily costs E1 in the intermediate state, the expansion is", + "\u0011 −2Jeff X ⟨ij⟩ ˆn[i+j]↑ˆn[i+j]↓ (27) where ˆn[i+j]σ ≡ \u0010 ˆc† iσ + ˆc† jσ \u0011 (ˆciσ + ˆcjσ) /2 is the number of electrons in a bonding orbital between sites i and j. Since any virtual movement of electrons necessarily costs E1 in the intermediate state, the expansion is valid as long as |t| ≪E1. [Note that, this condition does not require |t| ≪Ue-ph.] As can be seen from Table II, teffis second order in t and Jeffis fourth order. Thus, we should consider this theory in its weak coupling limit, Jeff≪teff. This interaction term on each bond can be decomposed into a sum of an AF coupling, a pair hopping term, and a density attraction. We analyze the weak-coupling instabilities in the context of a Hartree-Fock mean-field analysis that is reasonable in this limit. We find that, on square and triangular lattices, s-wave SC is always the dominant instability for n < 2. However, when n ≈1 on the square lattice, there is also a d-wave pairing state that is meta-stable. To understand the reason for this, one may rewrite the interaction in the form 1 Nsite P q,k,k′ Vq;kk′ˆc† k↓ˆc† q−k↑ˆcq−k′↑ˆck′↓. For q = 0 and for k, k′ near the Fermi surface at n ≈1, we obtain: Vkk′ ≈−2Jeff \u0002 2 + cos kx cos k′ x + cos ky cos k′ y \u0003 = −Jeff X a f (a)(k)f (a)(k′) (28) where in the factorized form, the sum runs over an s-wave channel (f (s)(k) = 2), an extended s-wave channel (f (¯s)(k) = cos kx + cos ky), and a d-wave channel (f (d)(k) = cos kx −cos ky). When n ≈1, where the system 10 has a large density of states near the two X points, (π, 0) and (0, π), of the Brillouin zone of the square lattice, the dx2−y2-wave channel with form factor f (d)(k) is only moderately subdominant to the dominant s-wave channel (which generically is a combination of s and ¯s). Moreover, the competition between the s- and d-wave paired states can be tuned by an additional weak on-site Hubbard repulsion, Uc; when Uc ≳2.2Jeff, the d-wave paired state has the lower variational energy. Exactly at n = 1, there is also an AF instability, which is also subdominant to the s-wave SC instability, but which is favored over all superconducting states when Uc > 2Jeff.", + "Skip to main content [image] We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate [image] Help | Advanced Search All fields Title Author Abstract Comments Journal reference ACM classification MSC classification Report number arXiv identifier DOI ORCID arXiv author ID Help pages Full text Search Login This version of 1503.04552v2 has been withdrawn and is unavailable See abstract page for more details • About • Help • Contact • Subscribe • Copyright • Privacy Policy • Web Accessibility Assistance • arXiv Operational Status Get status notifications via email or slack", + "Ultrafast Electrons in Noble Metals: Orientational Relaxation, Thermalization and Cooling in Terms of Electron-Phonon Interaction Jonas Grumm∗and Andreas Knorr† Nichtlineare Optik und Quantenelektronik, Institut f¨ur Physik und Astronomie, Technische Universit¨at Berlin, 10623 Berlin, Germany (Dated: April 29, 2025) We investigate the momentum-resolved dynamics of conduction electrons in noble metals following ultrashort optical excitation with linearly polarized light. Using a momentum-resolved Boltzmann equation approach for electron-phonon interaction, we solve for the combined effects of orientational relaxation, thermalization, and cooling. We introduce momentum orientational relaxation as the initial step in the equilibration of an optically excited non-equilibrium electron gas by highlighting its importance for the optical response of noble metals and the dephasing of plasmonic excitations. Our numerical results for gold reveal that orientational relaxation exists independently of the absorbed optical energy and dominates on time scales on the first tens of fs after excitation. Incorporating also thermalization and cooling on times up to a few ps, our approach provides a simultaneous description of optical and thermal properties of noble metals under initial non-equilibrium conditions. Introduction. The ultrashort excitation of electrons in noble metals by light fields with high intensities results in extreme non-equilibrium conditions. In the recent years, the development of such ultrashort excitations made it possible to observe of the dynamical optical response of solids with femtosecond resolution and provides new insights into the built-up and decay of non-equilibrium electronic or plasmonic excitations [1–6]. The relaxation back to a thermodynamic equilibrium, which proceeds in noble metals on scales from a few fs to several ps, depends strongly on the excitation conditions. The relaxation of the non-equilibrium energy-resolved electron distribu- tion itself can be divided in thermalization by electron- electron interaction [7], dissipation (electron cooling) by electron-phonon interaction [8–13] and lattice cooling by phonon-phonon interaction [14, 15]. These processes are addressed in many publications by effective macroscopic models [16–20] or microscopic theories based on the semi- classical energy-resolved Boltzmann equation [7–13, 21– 24]. These energy-resolved approaches neglect the fact that the polarization of the exciting light field breaks the sym- metry of the initially isotropic electron gas. This sym- metry break corresponds to an anisotropic excitation of electronic momentum which cannot be addressed in an energy-resolved basis only. However, the recent develop- ment of pulses short as several fs should make it possible to connect momentum-resolved measurements by using orthogonal polarized pump-probe experiments. In this Letter, we introduce the orientational relaxation of the momentum-polarized conduction electrons in noble metals as the first equilibrization step after optical exci- tation with linearly polarized light (cf. Fig. 1). We show that the orientational relaxation is fundamental for the optical response of noble metals [25], acts as dephasing of the collectively excited electrons (plasmons) in the for- mation of hot electrons, and effects the energy relaxation dynamics up to 100 fs after the excitation. So far, to our knowledge, momentum orientational relaxation was never discussed beyond the relaxation time approxima- tion [26–30] but occurs on the same footing with ther- malization and cooling of a non-equilibrium electron gas as a first dephasing process. To tackle this problem,", + "up to 100 fs after the excitation. So far, to our knowledge, momentum orientational relaxation was never discussed beyond the relaxation time approxima- tion [26–30] but occurs on the same footing with ther- malization and cooling of a non-equilibrium electron gas as a first dephasing process. To tackle this problem, we consider the numerical solution of a momentum-resolved Boltzmann equation exemplarily for gold including light- matter interaction and electron-phonon scattering [25]. Our approach includes orientational relaxation, thermal- ization as redistribution of electronic energy and cool- ing as energy dissipation from electrons to phonons. To study a well defined system, all three processes are de- scribed by a consistent treatment of the electron-phonon interaction via normal and umklapp processes. In the following, we will discuss these three relaxation processes in terms of the dynamics of the electron distribution and introduce characteristic time scales via observables. Theory. For the description of the dynamics of elec- trons in the sp-conduction band of noble metals after optical excitation, we apply a parabolic and isotropic dispersion ϵk = 1 2mℏ2|k|2 at their Fermi edge with an effective mass m at room temperature Teq = 300 K. For optical excitations below the electronic band gap, FIG. 1. A linear polarized light field E0 induces a momentum-polarization of the electron distribution fk creat- ing excited intraband electrons and holes near the Fermi edge. This state dominates on the first tens of fs after excitation. The excited non-symmetric distribution decays in the orienta- tional relaxation time τor into a symmetric distribution. The relaxation back into the thermodynamic equilibrium (TD eq.) proceeds in the thermalization and cooling time (τ ep th , τc) on longer time scales up to several ps. arXiv:2504.19745v1 [cond-mat.mes-hall] 28 Apr 2025 2 a description via intraband excitations is sufficient [29– 31]. Dissipative processes are incorporated via a screened electron-phonon coupling [8, 11, 32, 33]. By introducing the microscopic electronic conduction band occupations fk with momentum k, we obtain [25] the momentum- resolved Boltzmann equation [13, 15, 34] ∂tfk(t) = e ℏE0(t) · ∇kfk(t) + Γin k (t)(1 −fk(t)) −Γout k (t)fk(t) , (1) where Γin/out k (t) are the electron-phonon scattering ma- trices according to Fermi’s golden rule within a bath as- sumption in Born-Markov approximation as derived in Ref. [25]. Incorporating normal and umklapp processes, the scattering rates Γin/out k (t) taking into account the Pauli principle and include therefore an intrinsic many- body non-linearity. The excitation by a light field E0 acting on the electrons is described here as a vectorial quantity and not via the direction-independent intensity as in previous approaches focusing on energy relaxation out of a momentum-isotropic situation [7–13, 21–24]. In our approach, describing excitation and orientational re- laxation self-consistently, the light field breaks the sym- metry of the momentum distribution by the distinct po- larization direction of E0. Disturbing the momentum di- rection of the initial isotropic electron gas selectively, this opens the possibility to study orientational relaxation as new relaxation/dephasing channel. The Boltzmann equation (1) is solved numerically with gold parameters listed in Tab. 2 in Ref. [25]. In", + "by the distinct po- larization direction of E0. Disturbing the momentum di- rection of the initial isotropic electron gas selectively, this opens the possibility to study orientational relaxation as new relaxation/dephasing channel. The Boltzmann equation (1) is solved numerically with gold parameters listed in Tab. 2 in Ref. [25]. In the fol- lowing, we investigate the combined dynamics of electron relaxation in terms of orientational relaxation, thermal- ization and cooling after optical excitation in a bulk sys- tem (cf. Fig. 1). We apply a linearly y-polarized light field E0 represented by a single Gaussian pulse with a dura- tion of 12 fs (FWHM) and a carrier frequency of 5 THz. On the expected time scale of the relaxation dynamics described above, this pulse acts as an initial condition which allows to study the intrinsic electron relaxation without the influence of a finite pulse duration or fast os- cillating fields but rather the sequence of electron-phonon processes without interference of the initial excitation. In the following, we will study linear and non-linear exci- tation regimes. Both regimes can be distinguished via the distortion of the electron distribution, i.e. by the energy the electrons absorbed from the light field ϵabs (cf. Ref. [25]) in comparison with the thermal broaden- ing of the Fermi edge kBTeq: In the regime of non-linear excitations, the field strength is set to E0 = 0.7 MV cm (ϵabs ≳kBTeq), while for linear excitations E0/100 (ϵabs ≪kBTeq) is assumed. In this context, we ex- pect that orientational relaxation without any energy ex- change with the phonon system exists across all excita- tion regimes, while thermalization and cooling as energy relaxation processes occur only in the non-linear regime. Qualitative Discussion. The light-matter interaction in the Boltzmann equation (1) breaks the original ra- dial symmetry of the electron distribution in momentum space, since electrons are initially accelerated in the po- larization direction ey of the light field E0. Counter- acting to this, the electron-phonon interaction does not prefer any direction and thus starts to equilibrate all mo- mentum orientations to a distribution of uniform momen- tum directions. To analyze this process, we divide the electron distribution into a non-symmetric distribution f non−sym kx,ky,kz depending on the direction of the momentum k = (kx, ky, kz)T and a symmetric distribution f sym |k| (t) depending on the momentum absolute |k| fk(t) =f non−sym kx,ky,kz (t) + f sym |k| (t) , (2) f sym |k| (t) =f non−th |k| (t) + f th |k|(t) . (3) Later, to distinguish thermalization and cooling, the symmetric distribution f sym |k| is divided into thermal f th |k| and non-thermal f non−th |k| components [22, 35]. Thus, as illustrated in Fig. 1, orientational relaxation applies to the fact that the non-symmetric electron dis- tribution f non−sym kx,ky,kz induced by the light field E0 is driven into a symmetric distribution f sym |k| , which does not prefer a direction in momentum space. Our numerical results indicate that this process of momentum orientational re- laxation occurs independently of the optical excitation strength in the linear and non-linear", + "kx,ky,kz induced by the light field E0 is driven into a symmetric distribution f sym |k| , which does not prefer a direction in momentum space. Our numerical results indicate that this process of momentum orientational re- laxation occurs independently of the optical excitation strength in the linear and non-linear regime and without dissipating energy into the phonon bath. To illustrate this process, Fig. 2a and b depict the temporal occu- pation dynamics in the kx- (equivalent to kz), and ky- direction (orthogonal and parallel to the incident light polarization) for three different momentum states located below, at and above the Fermi edge. To enhance the vis- ibility of this process, we depict in Fig. 2 only the results for the non-linear excitation with strength E0 where the electron gas is strongly perturbed. Nevertheless in the linear regime, orientational relaxation shows the same qualitative signatures: Out of the initial equilibrium Fermi-Dirac distribution f eq k (Teq) during the first tens of fs after the excitation, the occupations in Fig. 2a clearly show a non-symmetric behavior of the distribution fk with respect to kx- and ky- direction. The onset of the dynamics in the kx-direction is temporally delayed compared to the ky-direction which is initially elongated by the light field on the scale of the pulse duration of 12 fs. After the orientational relax- ation time τor of less than 100 fs the directional depen- dence of fkx,ky,kz is temporally synchronized (cf. Fig. 2b), i.e. equilibrated with respect to the momentum direction and has developed into a symmetric distribution depend- ing only on |k|, i.e. on energy ϵk. Such an orientational relaxation has already been studied in the semi-metal graphene [36, 37], photoexcited semiconductors [38] or on the basis of a relaxation time approximation in met- als [28, 29], but as far as we know, has never been dis- cussed quantitatively for noble metals in terms of the full non-linear electron dynamics such as Eq. (1). In Fig. 2c and d, the evolution of the full electron distribution fk as function of ky and kx is visualized in time and momen- 3 delayed onset orientational relaxation 𝐄𝟎⊥𝐤𝒙 𝐄𝟎∥𝐤𝐲 FIG. 2. Dynamics of the electron gas after excitation with a y-polarized pulse with strength E0. In (a) the dynamics of the occupations in ky-direction (with kx = kz = 0, solid lines) and in kx-direction (with ky = kz = 0, dashed lines) are shown for different momentum states below, at and above the Fermi edge (dashed lines in c,d correspond to energies of EF and EF ± 2kBTeq, respectively). The decay of the occupation difference between ky- and kx-direction visualizes in (b) the orientational relaxation. The electron distribution in momentum space is depicted along the (c) ky- and (d) kx-axis (the kz- direction is identical with the kx-case) and represents a Fermi-Dirac distribution in equilibrium. tum space. Initially, before excitation at t = 0, c and d show a Fermi-Dirac distribution in equilibrium with the phonon bath and a Fermi level at kF = 12.0 nm−1 (dark areas describe occupied electron states and", + "is identical with the kx-case) and represents a Fermi-Dirac distribution in equilibrium. tum space. Initially, before excitation at t = 0, c and d show a Fermi-Dirac distribution in equilibrium with the phonon bath and a Fermi level at kF = 12.0 nm−1 (dark areas describe occupied electron states and bright areas unoccupied states). On the time scale of the excitation pulse in Fig. 2c, the strong displacement of the electron distribution parallel to the polarization of the light field in ky-direction is visible. While in the linear regime the electron gas is back in thermodynamic equilibrium after orientational relax- ation and no further relaxation proceeds, in the non- linear regime energy relaxation continues: Energy re- laxation includes thermalization as redistribution of the non-thermal electronic energy into thermal one and cool- ing as energy dissipation to the phonon bath [11, 22]. Thus, for non-linear excitations, orientational relaxation constitutes only the first temporal step in a sequence of relaxation events. The processes of thermalization and cooling as a function of energy are well studied in litera- ture [8–11, 22, 23]. Here, we would like to show that they develop in a consistent fashion out of the same theory as the orientational relaxation, i.e. by solving Eq. (1) in a momentum-resolved picture: After the orientational relaxation, electrons are dis- tributed symmetrically f sym |k| but not necessarily ther- mal (i.e. non-Fermi-Dirac), cf. Eq. (3), and an energy- dependent representation of the distribution in terms of ϵk = 1 2mℏ2|k|2 is permissible. In the next step, the transi- tion from a non-thermal distribution f non−th |k| into a ther- mal Fermi-Dirac distribution f th |k| due to electron-phonon scattering proceeds in the thermalization time τ ep th . At the same time, the excited hot electron gas dissipates its thermal energy to the phonon system in the cooling time τc. To distinguish between the simultaneously proceed- ing thermalization and cooling processes, we intro- duce the quasi-logarithmic representation Φsym |k| = −log ((f sym |k| )−1 −1) of the symmetric distribution as in Ref. [10]. This function connects a thermal Fermi-Dirac distribution linearly to the electron energy ϵk with slope proportional to the inverse electron temperature T −1 e . The solid lines in Fig. 3a illustrate our numerical re- sult for different times after excitation with the corre- sponding thermal distribution f th |k| as fitted Fermi-Dirac distribution (dashed lines). Their increasing slope over time consequently describes the cooling of the electron gas, whereas the decreasing deviation of the symmet- ric distribution from the thermal distribution describes thermalization (shaded areas). The complete relaxation of the electron gas from a hot distribution shortly after the excitation back to the equilibrium proceeds on times longer than 2.5 ps. For a more quantitative discussion of the time scales of orientational relaxation, thermalization and cooling, we connect the microscopic distributions in Eq. (2,3) to observables: a) Orientational relaxation. To discuss the orienta- tional relaxation time, we study the macroscopic current density j(t) = −2e V X k vkf non−sym kx,ky,kz (t) (4) defined via the electron velocity vk =", + "of orientational relaxation, thermalization and cooling, we connect the microscopic distributions in Eq. (2,3) to observables: a) Orientational relaxation. To discuss the orienta- tional relaxation time, we study the macroscopic current density j(t) = −2e V X k vkf non−sym kx,ky,kz (t) (4) defined via the electron velocity vk = 1 ℏ ∂ϵk ∂k . From symmetry considerations follows that the current den- 4 cooling thermalization FIG. 3. The quasi-logarithmic representation of the symmet- ric distribution Φsym |k| (solid lines) reveals the thermal (dashed lines) and non-thermal distribution (shaded areas). sity is carried exclusively by the non-symmetric distri- bution f non−sym kx,ky,kz [39]. This allows us to determine the orientational relaxation time directly from the temporal decay of the current density, Eq. (4), shown in Fig. 4 (yellow line and inset). In the regime of linear excita- tions with E0/100 (darkest line in the inset), the decay time corresponds to an orientational relaxation time of τor = 19.7 fs. In contrast, for non-linear excitations with field strength E0 (brightest line in the inset) the orien- tational relaxation time decreases to τor = 17.1 fs. This implies a reduction of the orientational relaxation time for stronger excitations, if the amount of absorbed en- ergy ϵabs is at least in the same order of magnitude as the thermal broadening kBTeq. The current density j acts as source term in Maxwell’s equations and is consequently essential for the optical re- sponse of the noble metal. As we report in Ref. [25], the orientational relaxation rate τ −1 or is in the linear regime identified with the damping rate γD, introduced phe- nomenologically in the classical Drude model [40, 41]. The Drude model assumes a field strength-independent damping in agreement with our observation for the orien- tational relaxation rate in the linear regime. Experimen- tal studies on the optical dielectric function of gold reveal damping times between 9 fs and 26 fs for excitations be- low the optical band gap [42–46]. Ab initio studies based on a relaxation time approximation predict values be- tween 24.0 fs and 26.3 fs [28, 30] in agreement with our results in the linear regime. b) Energy Relaxation. In the regime of linear exci- tations, the energy transfer by the light field to the elec- tron gas is negligible (ϵabs ≪kBTeq) and the definition of a thermalization and cooling time is unnecessary. In the non-linear regime, energy relaxation occurs for non- thermal and thermal electrons via an energetic redistri- bution and dissipation. To access the dynamics of non-thermal electrons, we de- fine the non-thermal energy density [10] unon−th e (t) = 1 π2 Z ∞ 0 dk k2ϵk(f sym |k| (t) −f th |k|(t)) . (5) 𝐸0 𝐸0/100 FIG. 4. The current density j decays exponentially within the orientational relaxation time τor. On longer time scales, the non-thermal energy density |unon−th| and effective tem- perature ⟨Te⟩track thermalization and cooling. The inset shows the decay of the current density for different excitation strengths from E0/100 (dark) to E0 (bright). The thermal distribution is addressed by a Fermi-Dirac function f th |k|(t) =", + "τor. On longer time scales, the non-thermal energy density |unon−th| and effective tem- perature ⟨Te⟩track thermalization and cooling. The inset shows the decay of the current density for different excitation strengths from E0/100 (dark) to E0 (bright). The thermal distribution is addressed by a Fermi-Dirac function f th |k|(t) = 1 exp n ϵk−⟨µ(t)⟩ kB⟨Te(t)⟩ o + 1 (6) where the effective temperature ⟨Te⟩and the chemical po- tential ⟨µ⟩are defined by the slope of the function Φsym |k| at the Fermi level (cf. Fig. 3). It is important to recog- nize that this effective temperature is not a function of state of the total electron gas but is defined exclusively for the thermal distribution and is only a measure for the thermal energy. As shown in Fig. 4, the thermalization, following on orientational relaxation, analyzed in terms of the non- thermal energy density reveals a more complex dynamics than a simple exponential decay. Here, a definition of a simple relaxation time is not possible but non-thermal electrons participate on a time span of τ ep th ≈2.0 ps in the relaxation dynamics. The decay and subsequent re-increase of the non-thermal energy |unon−th| between 0.1 ps and 0.8 ps after excitation suggests that electron- phonon scattering first leads to a state more distant from equilibrium before the final thermalization pro- ceeds. Similar observations have already been made for thermalization by electron-electron scattering [7]. The thermalization, described by our results exclusively in terms of electron-phonon interaction, is in agreement with observations in Ref. [10, 11]. Nevertheless, for a comprehensive description of thermalization, which is not the scope of this paper, electron-electron interaction has to be considered. Simultaneous to thermalization, the electron gas dissi- pates thermal energy to the phonon bath in a cooling time τc = 0.8 ps (red line in Fig. 4). Since heating and cooling of the electron gas are in a microscopic pic- ture intrinsically non-linear processes, the exact times depend on the strength of the light field and increase with increasing excitation strength. Similar excitation 5 dependent cooling times in gold are provided by versatile experimental and theoretical approaches between 0.8 ps and 1.2 ps [8, 16, 47–53]. Finally, the electron gas reaches the thermodynamic equi- librium after about 2.5 ps, when the non-thermal dis- tribution is decayed completely and the temperature ⟨Te⟩= Te = Teq becomes a state function of the full electron gas [22, 54]. Conclusion. We studied a momentum-resolved Boltz- mann equation for bulk gold to fill the gap between ultrafast optical excitation providing a momentum- polarized electron gas to its energy-resolved thermaliza- tion and cooling. During orientational relaxation, the non-symmetric electron distribution dominates for short times less than 100 fs directly after the excitation. After- wards for non-linear excitations during thermalization, the symmetric electron distribution develops out of a non-thermal into a thermal distribution and simultane- ously relaxes back to the equilibrium by dissipating en- ergy to the phonons in about 2.5 ps. Since all three relax- ation processes depend on the strength of the excitation, a model for the non-linear optical response of noble met-", + "out of a non-thermal into a thermal distribution and simultane- ously relaxes back to the equilibrium by dissipating en- ergy to the phonons in about 2.5 ps. Since all three relax- ation processes depend on the strength of the excitation, a model for the non-linear optical response of noble met- als must include many-body interactions such as Pauli blocking, shown here exemplarily for electron-phonon in- teraction. 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Vulpiani, Temperature in and out of equilibrium: A review of concepts, tools and attempts, Physics Reports Temperature in and out of equilibrium: a review of concepts, tools and attempts, 709-710, 1 (2017).", + "1 Electron transport properties of heterogeneous interfaces in solid electrolyte interphase on lithium metal anodes Xiangyi Zhou1*, Rongzhi Gao1, Ziyang Hu1,2*, Weijun Zhou3, YanHo Kwok3, GuanHua Chen1,2* 1Department of Chemistry, The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China 2Hong Kong Quantum AI Lab Limited, Pak Shek Kok, Hong Kong SAR, China 3QuantumFabless Limited, Pak Shek Kok, Hong Kong SAR, China *Corresponding authors. Email: XZ: xy.zhou2025@gmail.com, ZH: hzy@yangtze.hku.hk, and GHC: ghc@everest.hku.hk, ORCID: XZ: 0009-0002-8311-0768; RG: 0000-0001-7520-0347; ZH: 0000-0002- 7693-5457; WJZ: 0000-0002-4328-3704; YHK: 0000-0003-1325-4416; GHC: 0000- 0001-5015-0902 2 Abstract In rechargeable batteries, electron transport properties of inorganics in the solid- electrolyte interphase (SEI) critically determine the safety, lifespan and capacity loss of batteries. However, the electron transport properties of heterogeneous interfaces among different solid inorganics in SEI have not been studied experimentally or theoretically yet, although such heterogeneous interfaces exist inevitably. Here, by employing non- equilibrium Green’s function (NEGF) method, we theoretically evaluated the atomic- scale electron transport properties under bias voltage for LiF/Li2O interfaces and single- component layers of them, since LiF and Li2O are common stable inorganics in the SEI. We reveal that heterogeneous interfaces orthogonal to the external electric-field direction greatly impede electron transport in SEI, whereas heterogeneous parallel- orientated interfaces enhance it. Structural disorders induced by densely distributed interfaces can severely interfere with electron transport. For each component, single- crystal LiF is highly effective to block electron transport, with a critical thickness of 2.9 nm, much smaller than that of Li2O (19.0 nm). This study sheds a new light into direct and quantitative understanding of the electron transport properties of heterogeneous interfaces in SEI, which holds promise for the advancement of a new generation of high-performance batteries. Keywords: electron leakage; electron passivation; electron conductivity; non- equilibrium Green’s function (NEGF) method; critical thickness Introduction In rechargeable batteries with solid or liquid electrolyte, the electrolyte is positioned between the anode and the cathode to facilitate ion transport1. The Solid-Electrolyte Interphase (SEI) serves as a critical interface layer between the electrolyte and the 3 anode1-4. The characteristics of the SEI layer determine many key properties of rechargeable batteries5-8. The SEI can form naturally through self-limiting electrochemical decomposition of the electrolyte, or be introduced during the manufacturing process (known as artificial SEI)9,10. Naturally formed SEIs are multi- component, typically with inorganics near the electrode and organics near the electrolyte11,12. Among them, LiF and Li2O are two commonly stable inorganics in SEI13-15. Therefore, heterogeneous interfaces between different components in SEI inevitably exist and play a crucial role in determining the properties of the SEI and batteries14,15. However, the electron transport properties of heterogeneous interfaces among different solid inorganics in SEI are completely unknown to the best of our knowledge, despite of their importance in determining the safety and lifespan of batteries16,17. Electron transport across the SEI is one of the driving forces behind self-limiting electrolyte decomposition and dendrite formation3. An ideal SEI should have suitable thickness and be crack-free. Such an ideal SEI can block electron transport to protect the electrolyte and passivate the electrode, while still facilitating ion diffusion1,3. However, in", + "transport across the SEI is one of the driving forces behind self-limiting electrolyte decomposition and dendrite formation3. An ideal SEI should have suitable thickness and be crack-free. Such an ideal SEI can block electron transport to protect the electrolyte and passivate the electrode, while still facilitating ion diffusion1,3. However, in reality, electron transport across the SEI can still occur when it cracks or is insufficiently thick18. Currently, theoretical investigations of electron transport in the SEI are still in early stages. Multiple studies indirectly and qualitatively deduced the electron conductivity of the SEI layer by comparing electronic band structures19-21, density of states (DOS)19, or work function of different SEI components17. However, these studies cannot reflect the relationship between electron transport properties and applied bias, which is crucial given that SEIs were formed under voltage conditions22,23. In contrast, the non-equilibrium Green’s function (NEGF) method enables direct and quantitative analysis of the electron transport properties of SEI models under bias24-26. So far, only a few studies have employed the NEGF method to study the electron 4 transport properties of the SEI24,25. Benitez et al.25,26 evaluated the electron transport through several SEI components using cluster models in vacuum rather than dense solid layers. Such models of SEI components may not reflect the actual working conditions of the SEI. As an improvement, Smeu et al.24 used solid layers as structural models to investigate electron leakage through grain boundaries inside LiF on lithium–metal anodes. However, research on the electron transport properties of interfaces between different components in SEI remains lacking. Therefore, we adopted structural models similar to those used by Smeu et al. to explore the atomic-scale electron transport properties of heterogeneous interfaces between different components in the SEI on lithium metal anodes. We employed the NEGF method to study the current density under bias voltage for LiF and Li2O layers, as well as the interfaces formed by these two components. The remaining part of this paper is organized as follows. First, through fitting the relationship between thickness and current density at 2 V, we predicted the critical thicknesses of single-crystal LiF and Li2O needed to effectively block electron transport in SEI. Then, we investigated electron transport properties of heterogeneous interfaces orthogonal to the electric-field direction by analyzing four systems with orthogonal LiF/Li2O interfaces and single-component LiF and Li2O systems. Next, we explored electron transport properties of heterogeneous interfaces parallel to the electric-field direction by studying four systems with parallel LiF/Li2O interfaces. This work reveals the contrasting effects of heterogeneous SEI interfaces orthogonal and parallel to the external electric field direction, and may help the rational design of artificial SEI and improve the performance of rechargeable batteries. 5 Results and discussion We conducted atomic-scale electron transport simulations using the SEI systems illustrated in Fig. 1. An SEI layer containing selected combinations of Li2O and LiF solids is placed between two semi-infinite lithium electrodes which are assumed to be in equilibrium with potentials μL and μR, respectively, as shown in Fig. 1a. For studied systems with LiF/Li2O interfaces, the thickness of the simulated SEI layers along the electric field", + "containing selected combinations of Li2O and LiF solids is placed between two semi-infinite lithium electrodes which are assumed to be in equilibrium with potentials μL and μR, respectively, as shown in Fig. 1a. For studied systems with LiF/Li2O interfaces, the thickness of the simulated SEI layers along the electric field direction is around 1.6 nm, as shown in Fig. 1b. To make fair comparison between the conductivity of different SEI systems and to predict the critical thickness, simulations have first been performed on pure LiF and Li2O with various thicknesses. These SEI layers were initially built with atomic layers of LiF(001) crystal plane and/or Li2O(111) crystal plane stacked along the external electric field direction, sandwiched between two lithium leads. In the other two directions orthogonal to the external field direction, the systems are assumed to be periodic, with a unit cell size of approximately 1.7 nm × 1.7 nm, as shown in Fig. 1c. Predicting critical thickness of LiF and Li2O When the thickness reaches a critical value (i.e., the critical thickness), the tunneling probability and corresponding electric current density (i.e., the critical current density) are so low that the growth of SEI becomes negligible. To understand the influence of thickness on electron transport properties and to predict the critical thickness of single- crystal LiF and Li2O, we analyzed the steady-state current density of LiF and Li2O with different thickness under bias voltages ranging from 0V to 2V. Specifically, we investigated four LiF layers with thicknesses of 0.8, 1.6, 2.4, and 3.2 nm, and six Li2O layers with thicknesses of 0.8, 1.6, 2.4, 3.2, 6.4, and 12.8 nm. Structural schematics of LiF-1.6 nm and Li2O-1.6 nm systems are already shown in Fig. 2a-b, other structures are shown in Fig S2. Fig. 3a and b show I-V relationship for LiF and Li2O with different 6 thicknesses, respectively. LiF systems consistently demonstrated current densities at least four orders of magnitude lower than equivalent Li2O system. This highlights LiF’s superior effectiveness in blocking electron transport across the SEI compared to Li2O, consistent with LiF being commonly recognized as a preferable candidate for an artificial SEI over Li2O27-29. This contrasts with another NEGF study by Benitez et al.25, where LiF’s resistance is similar to Li2O’s at lower bias voltages, with differences emerging only at higher bias voltages (> 4 V). The inconsistency in simulation results may stem from SEI layers being modeled as clusters in the vacuum in their study but as solid layers in ours. Then, through fitting the relationship between thickness and current density at 2 V, we predicted the critical thickness of LiF and Li2O, as shown in Fig. 3c and d, respectively. The logarithm of total current density ( j ) showed a robust linear relationship with SEI thickness(d): for LiF, the relationship was described by j/(mA cm-2)=10-5.21d/(nm)+12.97 with an R² greater than 0.99, while for Li2O, it was approximately 0.98, described by j/(mA cm-2)=10-0.76d/(nm)+12.67. The larger magnitude of the coefficient in front of d for LiF (–5.21) compared to Li2O (–0.76) indicates that LiF exhibits a higher resistivity. Furthermore, taking 0.01", + "relationship was described by j/(mA cm-2)=10-5.21d/(nm)+12.97 with an R² greater than 0.99, while for Li2O, it was approximately 0.98, described by j/(mA cm-2)=10-0.76d/(nm)+12.67. The larger magnitude of the coefficient in front of d for LiF (–5.21) compared to Li2O (–0.76) indicates that LiF exhibits a higher resistivity. Furthermore, taking 0.01 mA/cm² as the critical current density30, the corresponding critical thicknesses for LiF and Li2O are around 2.9 nm and 19.0 nm, respectively. This significant difference in critical thickness again underscores LiF’s superiority in blocking electron transport compared to Li2O. Electron transport properties of heterogeneous orthogonal interfaces To explore the electron transport properties of heterogeneous interfaces orthogonal to the external electric field induced by bias voltage, we analyzed four systems featuring such orthogonal LiF/Li2O interfaces, alongside single-component LiF and Li2O systems for comparison. The structural configurations of these systems are depicted in 7 Fig. 2a-f and Fig. S2. The four systems with LiF/Li2O interfaces (depicted in Fig. 2c-f) feature structures containing the same number of each type of atoms. Specifically, these four simulated SEI layers consist of 4 Li/F atomic layers for LiF and 6 Li atomic layers plus 3 O atomic layers for Li2O. For comparison, we also studied single-component LiF and Li2O systems with thicknesses of both 0.8 nm (Fig. S2) and 1.6 nm (Fig. 2a-b). The 0.8 nm SEI layers consist solely of LiF (4 Li/F layers) and Li2O (6 Li plus 3 O layers), matching the component counts found in the heterogeneous interface structures. For the 1.6-nm single-component systems, they contain twice the number of LiF/Li2O layers while maintaining similar total thickness when compared to the four heterogeneous interface systems. The interfaces in these heterogeneous systems are different, detailed as follows: • The LiF/Li2O system (Fig. 2c) includes a Li/LiF interface, a LiF/Li2O interface, and a Li2O/Li interface, all orthogonal to the electric field direction. • The LiF/Li2O/LiF/Li2O system (Fig. 2d) contains a Li/LiF interface, three LiF/Li2O interfaces, and a Li2O/Li interface, • The LiF/Li2O/LiF system (Fig. 2e) includes two Li/LiF interfaces and two LiF/Li2O interfaces. • The Li2O/LiF/Li2O system (Fig. 2f) contains two LiF/Li2O interfaces and two Li2O/Li interfaces. For all four systems with LiF/Li2O interfaces, the structural configuration shows only slight distortion near the interfaces after optimization, indicating a favorable structural match between the LiF(001) and Li2O(111) crystal planes stacked along the electric field direction. Fig. 4a illustrates the steady-state current density of these systems under bias voltages ranging from 0V to 2V. First, we compared the above four interface systems 8 with single-component LiF and Li2O systems. Neglecting the influence of orthogonal LiF/ Li2O interfaces, the linear relationship established in the previous section suggests that the four systems with orthogonal interfaces would exhibit a current density of approximately 1.70×108 to 2.97×108 mA/cm2 at 2V, due to the similar total thickness of LiF (~0.8 nm) and Li2O (~0.8 nm) layers in these systems. However, the actual current densities are significantly lower, ranging from 4.62×106 to 7.05×107 mA/cm2. This discrepancy indicates that the orthogonal heterogeneous interfaces impede electron transport. We further compared the electron transport properties across the", + "the similar total thickness of LiF (~0.8 nm) and Li2O (~0.8 nm) layers in these systems. However, the actual current densities are significantly lower, ranging from 4.62×106 to 7.05×107 mA/cm2. This discrepancy indicates that the orthogonal heterogeneous interfaces impede electron transport. We further compared the electron transport properties across the four configurations. Notably, the current density of the Li2O/LiF/Li2O/LiF system was found to be lower than that of the LiF/Li2O system across the bias range from 0V to 2V. This distinction underscores the effectiveness of the orthogonal LiF/Li2O interface in blocking electron transport, attributed to the former system’s inclusion of three LiF/Li2O interfaces compared to just one in the latter. To delve deeper into their electron transport characteristics and identify dominant energy ranges at a 2V bias, we analyzed transmission and current density spectra for both systems, as shown in Fig. S3a-b. The transmission spectrum highlights the energy ranges of the main peaks, indicative of the most conductive channels for electron transport. The current spectrum delineates energy channels contributing significantly to the overall current, with the integral area under the curves corresponding to the current at 2 V bias. Within a wide range around the Fermi energy from –0.9 eV to 0.9 eV, the LiF/Li2O system exhibits a larger transmission coefficient, facilitating wider electron transport. This difference is further illustrated in the current spectrum, where the total current is primarily contributed by energy channels between –1.1 eV to 1.1 eV for both systems. Importantly, the area under the current spectrum curve is significantly larger for the LiF/Li2O system compared to the LiF/Li2O/LiF/Li2O system, confirming the superior electron transport- 9 blocking capability of the orthogonal LiF/Li2O interface. It should also be noted that the current density of the LiF/Li2O/LiF system is approximately an order of magnitude lower than that of the Li2O/LiF/Li2O system. This highlights the superior electron-blocking capability of the orthogonal Li/LiF interface compared to the orthogonal Li/Li2O interface, as these two systems contain the same amount of LiF and Li2O layers. It is because the main structural difference between these two systems lies in the number and type of interfaces: the LiF/Li2O/LiF system features two Li/LiF interfaces, whereas the Li2O/LiF/Li2O system features two Li/Li2O interfaces. Transmission and current density spectra for both systems are provided in Fig. S3c-d. Near the bias window range, the most conductive energy channels for electron transport in the LiF/Li2O/LiF system are confined narrowly from –1.4 eV to – 1.3 eV, whereas those in the Li2O/LiF/Li2O system are broadly distributed from –1.4 eV to 1.4 eV, with significantly higher values than the LiF/Li2O/LiF system spanning from –1.3 eV to 1.4 eV. Consequently, the Li2O/LiF/Li2O system exhibits a notably higher total probability for electron transport. This difference is further emphasized in the current spectrum shown in Fig. S3d, where the current density for Li2O/LiF/Li2O consistently exceeds that of LiF/Li2O/LiF by more than an order of magnitude across the entire energy range. This analysis confirms the effectiveness of the orthogonal Li/LiF interface in inhibiting electron transport compared to its Li/Li2O counterpart. Electron transport properties of parallel interfaces To explore", + "where the current density for Li2O/LiF/Li2O consistently exceeds that of LiF/Li2O/LiF by more than an order of magnitude across the entire energy range. This analysis confirms the effectiveness of the orthogonal Li/LiF interface in inhibiting electron transport compared to its Li/Li2O counterpart. Electron transport properties of parallel interfaces To explore electron transport properties across heterogeneous interfaces aligned parallel to the external electric field induced by bias voltage, we analysed four systems featuring parallel LiF/Li2O interfaces, alongside single-component LiF and Li2O systems for comparison. The structural configurations of these systems are illustrated in Fig. 2. The simulated SEI layers in these systems comprise varying ratios of LiF and Li2O, with 10 differing degrees of disorder. Specifically, in each periodic cell, the Li2O-rich system in Fig. 2g consists of 4 Li/F atomic layers and 11 Li/O atomic layers; the Li2O-LiF- balanced system in Fig. 2h consists of 6 Li/F atomic layers and 8 Li/O atomic layers; and the LiF-rich system in Fig. 2i consists of 8 Li/F atomic layers and 5 Li/O atomic layers. Fig. 2j shows the twice-interface system with the same total number of atomic layers as the balanced system, but with interleaved atomic layers resulting in twice as many interfaces parallel to the electric field direction in each unit cell. In each periodic cell, the 1.6 nm single-component LiF system in Fig. 2a contains 12 Li/F atomic layers aligned parallel to the electric field, while the 1.6 nm single- component Li2O system in Fig. 2b contains 18 Li/O atomic layers similarly aligned. In each periodic cell, the interfaces in these heterogeneous systems vary as detailed below: • The Li2O-rich system (Fig. 2g) features two LiF/Li2O interfaces parallel to the electric field direction, with Li2O thickness comprising roughly two thirds of the total thickness. • The Li2O-LiF-balanced system (Fig. 2h) includes two LiF/Li2O interfaces, with approximately equal thicknesses for LiF and Li2O. • The LiF-rich system (Fig. 2i) contains two LiF/Li2O interfaces, with Li2O thickness comprising roughly one third of the total thickness. • The twice-interface system (Fig. 2j) incorporates four LiF/Li2O interfaces, each with roughly equal distances (approximately one fourth of the total thickness) between adjacent interfaces. The reduced thickness of each layer and the increased number of interfaces lead to a much more disordered structure compared to the other three. These structures exhibit greater distortion near the LiF/Li2O interfaces compared 11 to the orthogonal cases, indicating a less ideal match between the LiF(001) and Li2O(111) layers stacked parallel to the external field direction, possibly due to the formation of Li-LiF-Li2O triple phase boundaries. Fig. 4b illustrates the steady-state current density of these systems under bias voltages ranging from 0 V to 2 V. First, we compared four systems featuring LiF/Li2O interfaces with single-component LiF and Li2O systems. The current density of four systems with LiF/Li2O interfaces is substantially lower than that of Li2O at 1.6 nm, but significantly higher than that of LiF at 1.6 nm. This indicates that electron transport in structures with parallel interfaces is predominantly determined by the component with higher electron transport capability. In Li2O-rich, Li2O-LiF-balanced,", + "of four systems with LiF/Li2O interfaces is substantially lower than that of Li2O at 1.6 nm, but significantly higher than that of LiF at 1.6 nm. This indicates that electron transport in structures with parallel interfaces is predominantly determined by the component with higher electron transport capability. In Li2O-rich, Li2O-LiF-balanced, and LiF-rich systems, the current density decreases with reduced Li2O content, reflecting Li2O’s superior conductivity. As shown in Fig. S4a, transmission spectra at 2 V reveals a conductive energy range from –1.4 eV to –0.6 eV for pure Li2O, Li2O-rich, and Li2O- LiF-balanced systems. LiF-rich system exhibits the most conductive energy range narrowing to –1.4 eV to –1.3 eV, with transmission levels in this range lower by approximately an order of magnitude compared to the other systems. Furthermore, the transmission spectra in Fig. S4c confirmed that the pure LiF system has significantly lower transmission levels compared to all other systems, with values at least four orders of magnitude smaller. Surprisingly, above 1.5 V, the Li2O-rich system exhibits current densities comparable to or higher than pure Li2O, indicating enhanced electron transport through the SEI layer via parallel interfaces, contrary to findings from orthogonal systems. As shown in the current spectrum in Fig. S4b, the Li2O-rich system is more conductive than the pure Li2O from –0.6 to –0.9 eV, which constitutes the main contribution to the electric current. This explains the higher current density of the Li2O-rich system and highlights the electron transport enhancement by parallel interfaces. This observation 12 is also consistent with the NEGF study by Smeu et. al.24, where a similarly oriented boundary between LiF grains of different crystal orientations enhanced the leakage current. It should also be noted that the current density of the twice-interface system is more than an order of magnitude lower than that of the Li2O-LiF-balanced system, despite containing the same total amount of LiF and Li2O. The increase in interface density leads to a higher degree of structural disorder, which significantly disrupts electron transport. This is evident when comparing the two systems: the Li2O-LiF- balanced system, with its two LiF/Li2O interfaces and four triple phase boundaries per unit cell, contrasts sharply with the twice-interface system, which has four interfaces and eight triple phase boundaries. As depicted in the transmission spectrum in Fig. S4e, for the Li2O-LiF-balanced system, the most conductive energy channels for electron transport span from –1.4 eV to –0.5 eV, narrowing to –1.4 eV to –0.8 eV as the number of interfaces is doubled, accompanied by a noticeable decrease in transmission. This difference is further illustrated in the current spectrum in Fig. S4f, where the twice- interface system shows a four-fold decrease. This analysis confirms the disruptive role of structural disorder in electron transport across the entire conductive energy range. The insights gained from our study provide a profound understanding of the phenomena occurring in rechargeable batteries from the perspective of electron transport. We have found that orthogonal interfaces and structural disorders act as impediments to electron transport, thereby enhancing electron-blockage and passivation effects. Conversely, parallel interfaces exhibit a different behavior. This implies that", + "our study provide a profound understanding of the phenomena occurring in rechargeable batteries from the perspective of electron transport. We have found that orthogonal interfaces and structural disorders act as impediments to electron transport, thereby enhancing electron-blockage and passivation effects. Conversely, parallel interfaces exhibit a different behavior. This implies that in the design of artificial SEIs, thinner layers with a higher prevalence of orthogonal interfaces and structural disorders can deliver comparable passivation benefits as thicker layers with a greater presence of parallel interfaces. Furthermore, our findings suggest that during the cycling of rechargeable batteries, dendrite growth 13 may be slower at orthogonal interfaces and locations with structural disorders compared to regions with parallel interfaces. Conclusions In summary, we employed the NEGF method to theoretically investigate atomic-scale electron transport properties under bias for LiF/Li2O interfaces and single-component atomic layers of these two components. We predicted that the critical thicknesses for effectively blocking electron transport in single-crystal LiF and Li2O would be 2.9 nm and 19.0 nm, respectively. Furthermore, we found that LiF/Li2O interfaces orthogonal to the electric-field direction impede electron transport. Conversely, LiF/Li2O interfaces parallel to the external electric-field direction can enhance electron transport but structural disorder induced by densely distributed interfaces severely interferes with electron transport. This study provides valuable insights for rational design of effective artificial SEI layers, and may further lay the foundation for improving the performance of rechargeable batteries. Methodology Non-equilibrium Green’s function for studying electron transport The electron transport properties of the systems are studied with the non-equilibrium Green’s function (NEGF) method. The key construction in NEGF is the retarded Green’s function of the reduced system comprising the central region, and it can be calculated using the following formula, 𝐺(𝐸) = [𝐸𝑆−𝐻−𝛴L(𝐸) −𝛴R(𝐸)−1] (1) where 𝑆 and 𝐻 stand for the overlap matrix and the Hamiltonian of the reduced 14 system. The 𝛴L and 𝛴R are the self-energies matrices for the left and right electrodes, respectively. They are complex energy-dependent matrices and account for the influence of the electrodes on the central region. The real parts of the self-energies account for the shift of the energy levels. The imaginary parts, 𝛤L,R = 𝑖(𝛴L,R − 𝛴L,R † ), are known as the line-width function and account for the broadening. The self-energies are evaluated iteratively from the electrode and device Hamiltonians. The expression for the electron density matrix is as follows, 𝜌= 1 2𝜋∫ [𝑓(𝐸, 𝜇L)𝐺𝛤L𝐺† + 𝑓(𝐸, 𝜇R)𝐺𝛤R𝐺†]d𝐸 ∞ −∞ (2) where the Fermi distribution functions 𝑓(𝐸, 𝜇L) and 𝑓(𝐸, 𝜇R) give the occupancy of the electron orbital at energy 𝐸 for an electrode at equilibrium with electrochemical potential 𝜇L,R. The difference between the two electrochemical potentials gives the bias voltage 𝑒𝑉b = 𝜇L − 𝜇R . The electron density matrix calculated above is used to update the local potential distribution, which is in turn used to update the Hamiltonian and the retarded Green’s function. This process is proceeded iteratively until self- consistency is achieved. The steady-state transmission function is then evaluated using the converged Green’s function as 𝑇(𝐸, 𝑉b) = Tr(𝛤L𝐺𝛤R𝐺†) (3) which describes the transmission probability of", + "potential distribution, which is in turn used to update the Hamiltonian and the retarded Green’s function. This process is proceeded iteratively until self- consistency is achieved. The steady-state transmission function is then evaluated using the converged Green’s function as 𝑇(𝐸, 𝑉b) = Tr(𝛤L𝐺𝛤R𝐺†) (3) which describes the transmission probability of an electron to transport between the electrodes via a channel with energy 𝐸 in the central region. The product of the transmission function and the difference of the Fermi functions of the two electrodes d𝐼 d𝐸 = 2𝑒 ℎ𝑇(𝐸, 𝑉b)[𝑓(𝐸, 𝜇L) − 𝑓(𝐸, 𝜇R)] is known as the current spectrum. It indicates the contribution to the total current from the specific energy channel. Its integration 15 gives back the Landauer-Büttiker formula for total current31, 𝐼(𝑉b) = 2𝑒 ℎ∫ 𝑇(𝐸, 𝑉b)[𝑓(𝐸, 𝜇L) − 𝑓(𝐸, 𝜇R) ∞ −∞ ]d𝐸 (4) The transmission per unit area, current spectrum per unit area, and current density (j) are obtained through dividing the area of the interfaces. Computational methods Our simulation integrates the DFT method for structural optimization with the NEGF- DFTB method for electronic transport property calculations. The DFT method provides relatively precise descriptions of SEI layer structures, whereas the DFTB method, known for its efficiency, enables NEGF simulations of SEI layers with significantly greater thicknesses32,33. The method of building structures for these two steps is illustrated in Fig. S1, and VESTA is employed for visualization of all structures. The Vienna Ab initio Simulation Package (VASP, version 5.4.4) is used for structural relaxation. The Perdew–Burke–Ernzerhof34 (PBE) approximation is used for exchange-correlation energy. The projector augmented wave (PAW) method handles electron interactions using a plane-wave basis expanded to 500 eV35-37. The Monkhorst–Pack scheme samples the 1 × 1 × 1 Brillouin zone, and Gaussian smearing (0.05 eV) is used as the occupation method. The electronic self-consistent convergence criterion is 10−5 eV. The electron transport calculations are performed with LODESTAR38. DFTB parameters are generated using our own software by fitting electronic structures of various solid systems containing Li, F, and O atoms against DFT results at PBE level 39,40. The structure for transport simulation is constructed by replicating two layers of 16 fixed lithium atoms from the optimized structure on both sides. Initially, DFTB Mulliken charges for the electrodes are derived from simulations on infinite equilibrium systems. These charges are subsequently used in a self-consistent process to solve the Poisson equation and NEGF equations for the entire system under specific bias conditions, yielding Mulliken charges for the device region. Finally, electronic transport properties, including transmission coefficients and current spectra, are computed through integration following the NEGF scheme. Data availability The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. References 1 Xu, K. Electrolytes and Interphases in Li-Ion Batteries and Beyond. Chemical Reviews 114, 11503-11618 (2014). https://doi.org/10.1021/cr500003w 2 Peled, E. The Electrochemical Behavior of Alkali and Alkaline Earth Metals in Nonaqueous Battery Systems—The Solid Electrolyte Interphase Model. J. Electrochem. Soc. 126, 2047 (1979). https://doi.org/10.1149/1.2128859 3 Wang, A., Kadam, S., Li, H., Shi, S. & Qi,", + "Interphases in Li-Ion Batteries and Beyond. Chemical Reviews 114, 11503-11618 (2014). https://doi.org/10.1021/cr500003w 2 Peled, E. The Electrochemical Behavior of Alkali and Alkaline Earth Metals in Nonaqueous Battery Systems—The Solid Electrolyte Interphase Model. J. Electrochem. Soc. 126, 2047 (1979). https://doi.org/10.1149/1.2128859 3 Wang, A., Kadam, S., Li, H., Shi, S. & Qi, Y. 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Self-Consistent-Charge Density-Functional Tight- Binding Parameters for Modeling an All-Solid-State Lithium Battery. Journal of Chemical Theory and Computation (2023). https://doi.org/10.1021/acs.jctc.2c01115 40 Porezag, D., Frauenheim, T., Köhler, T., Seifert, G. & Kaschner, R. Construction of tight-binding-like potentials on the basis of density-functional theory: Application to carbon. Physical Review B 51, 12947-12957 (1995). https://doi.org/10.1103/PhysRevB.51.12947 21 Acknowledgements GHC acknowledges financial support by the General Research Fund (Grant No. 17309620) and Research Grants Council (RGC: T23-713/22-R). GHC acknowledges support from the Hong Kong Quantum AI Lab, AIR@InnoHK of the Hong Kong Government. Competing interests The authors declare no competing interests. 22 Figures Fig. 1. Schematics of the SEI systems for simulating electron transport properties. a, Schematic of the whole circuit. The SEI layer (blue) is placed between two lithium electrodes (green), where the external bias voltage is applied. b, Schematic of atomic configuration of SEI systems for NEGF simulation, demonstrated by the side-view of LiF-1.6 nm system. The SEI layer is indicated by the dashed blue box. For all studied systems with LiF/Li2O interfaces, the thicknesses of SEI layers are around 1.6 nm. The direction of external electric field induced by bias voltage goes from one electrode to the other, as indicated by the blue arrow. c, Cross-section of a unit cell of the system in (b), for demonstrating the periodicity and size of studied systems. All studied systems are periodic in directions orthogonal to the electric field direction, with a unit cell length around 1.7 nm. 23 24 Fig. 2. Structures for studied system with SEI layers around 1.6 nm. a, the LiF-1.6 nm system. b, the Li2O-1.6 nm system. c-f show four systems with heterogeneous SEI interfaces orthogonal to the electric field direction, while g-j show four systems with parallel-oriented SEI interfaces. c, the LiF/Li2O system, containing a LiF layer and a Li2O layer of roughly equal thickness. d, the LiF/Li2O/LiF/Li2O system, containing the same amount for each component as in (c), but twice as many heterogeneous interfaces. e, the LiF/Li2O/LiF system, with two LiF/Li interfaces. f, the Li2O/LiF/Li2O system, with two Li2O/Li interfaces. g, the Li2O-rich system, comprising a thicker layer of Li2O than LiF. h, the Li2O-LiF-balanced system, comprising a layer of Li2O and a layer of LiF of comparable thicknesses. i, the LiF-rich system, comprising a thicker layer of LiF than Li2O. j, the twice-interface system, comprising the same amount of Li2O and LiF as (h) but with denser interface population. The blue dashed box in each structure indicates the SEI layer with thickness around 1.6nm. 25 Fig. 3. Relation between electronic current density and SEI layer thickness, highlighting the critical thicknesses. a, I-V curves for pure LiF SEI layer with thicknesses 0.8, 1.6, 2.4, and 3.2 nm. b, I-V curves for pure Li2O SEI layer with thicknesses 0.8, 1.6, 2.4,", + "layer with thickness around 1.6nm. 25 Fig. 3. Relation between electronic current density and SEI layer thickness, highlighting the critical thicknesses. a, I-V curves for pure LiF SEI layer with thicknesses 0.8, 1.6, 2.4, and 3.2 nm. b, I-V curves for pure Li2O SEI layer with thicknesses 0.8, 1.6, 2.4, 3.2, 6.4, and 12.8 nm. c, correlation analysis of current density at 2V against thickness for LiF systems, showing the critical thickness 2.9 nm corresponding to critical current density 0.01 mA/cm2. d, correlation analysis of current density at 2 V against thickness for Li2O systems, showing the critical thickness 19.0 nm corresponding to critical current density 0.01 mA/cm2. In (c, d), the variables d and j represent the thickness (nm) of the studied single component and the current density (mA/cm2) of the investigated systems at a voltage of 2 V, respectively. Fig. 4. I-V curves for studied systems with heterogeneous SEI interfaces under a bias range of 0~2V. a and (b) show orthogonal and parallel-oriented interfaces systems, respectively, with single-component systems included for comparison. (a) indicates that heterogeneous interfaces orthogonal to the electric field direction leads to a decreased current density. (b) shows the lower current density of the twice-interface system (shown in Fig. 2j) than Li2O-LiF-balanced system the system (shown in Fig. 2h) as denser interfaces are included in the former system. The grey dashed box highlights the 26 crossing of current between the Li2O-rich system (shown in Fig. 2g) and Li2O-1.6 nm system at high bias voltage. Graphical abstract", + "arXiv:cond-mat/9909304v1 21 Sep 1999 Electron Locking In Semiconductor Superlattices F.V.Kusmartsev Department of Physics, Loughborough University, LE11 3TU, UK and Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801 Landau Institute, Moscow, Russia Harjinder Singh Dhillon Department of Physics, Loughborough University, LE11 3TU, UK (September 13, 2018) Abstract We describe a novel state of electrons and phonons arising in semicon- ductor superlattices (SSL) due to strong electron-phonon interactions. These states are characterized by a localization of phonons and a self-trapping or locking of electrons in one or several quantum wells due to additional, de- formational potential arising around these locking wells in SSL. The effect is enhanced in a longitudinal magnetic field. Using the tight-binding and adiabatic approximations the whole energy spectrum of the self-trapped states is found and accurate, analytic expressions are included for strong electron-phonon coupling. Finally, we discuss possible experiments which may detect these predicted self-trapped states. PACS numbers: 63.20.Kr, 74.80.Dm, 73.20.Dx, 79.60.Jv Modern technology allows materials and structures with narrow bandwiths to be created. One example of such a structure is the semiconductor superlattice which can be manufac- tured in such a way that it can consist of different numbers of layers of differing thicknesses. In SSL consisting of say, for example, a sequence of layers of GaAs and GaxAl1−xAs, an electron miniband of width t may be created, where t is the overlapping integral for an elec- tron localized in neighboring quantum wells. The number of levels in this miniband depend on the number of layers the SSL consists of. For example, if the superlattice consists of two quantum wells, there are only two levels associated with the symmetrical and antisymmetric wave functions. For each energy level, the probability to find the electron in either well is equally distributed. The same is true when there are many levels in the miniband. However, the situation changes dramatically if electron-phonon interactions are considered. The effect of this interaction may give rise to coupled electrons and phonons where the dimensionless electron-phonon coupling constant of this system is inversely proportional to t. Since the miniband width, in SSL, is between 1 - 100 meV the electron-phonon coupling is strongly 1 intensified in comparison with the conventional bandwidth which is about 1 eV. Such an intensification in the electron-phonon interaction in narrow band structures may give rise to some novel, self-trapped states of the electron. This phenomenon of the self-trapping of electrons and excitons in solids, originally pre- dicted theoretically1–5, has been observed and well studied in many materials (see, for exam- ple, review4). It arises, primarily, in low dimensional systems and in systems with a strong electron-phonon interaction. The SSL is a new system where the effective electron-phonon coupling may become large, especially when a longitudinal magnetic field is applied. This can give rise to the locking of an electron in some wells with the creation of novel self-trapped states. The number of such states depend on how many layers the SSL consists of and how strong the electron- phonon interactions are, ie the narrowness of", + "magnetic field is applied. This can give rise to the locking of an electron in some wells with the creation of novel self-trapped states. The number of such states depend on how many layers the SSL consists of and how strong the electron- phonon interactions are, ie the narrowness of the miniband. The structure of these states is associated with a localization pattern. This pattern indicates where the electron is localized, ie, in which wells the electron is locked in. The system can be described in the framework of a 1D tight-binding model. If the interaction of the electron with acoustic phonons is considered then the Schr¨odinger equation takes the form: −∆ψn 2m⊥ −tψn−1 + 2tψn −tψn+1 + DQnψn = Eψn (1) where ∆= ∂2/∂x2 + ∂2/∂y2 is a two dimensional Laplacian associated with the electron motion inside the planes of the quantum wells, m⊥is the effective, transverse mass of the electron, ψn(x, y) is the wave function for an electron located in the nth quantum well and D is a constant of the deformation potential which depends on the material the SSL is made up of. Note that deformations along all 3 coordinate axes have been included because Qn = ∇.u = ∂xux+∂yuy+∂zuz where un = (unx, uny, unz) is a vector of atomic displacements in the nth quantum well. Thus, Qn includes deformations in the x, y and z directions. Using the adiabatic approximation, in which the lattice moves slowly in comparison with the electrons (because of the large mass of atomes Ma and small, effective mass me = ¯h2/ta2 of the electrons), the kinetic energy of the lattice with the parameter me/Ma →0 is zero. The adiabatic self-trapped states are associated with the stationary points of the adiabatic potential. In other words employing the adiabatic parameter (the ratio of electron and atomic masses me/Ma << 1) following Pekar1 we consider different static deformations of the lattice and the localized electron states created by these deformations. Extreme (or critical) points of this adiabatic surface are self-trapped states. Traditionally, adiabatic self-trapping has only been studied in continuum models5 appli- cable to localized states with a large radius (in comparison to the lattice spacing), except, of course, the exciton self-trapping in rare gas solids4. The same is true for studies of the interaction of an electron with polar phonons (polarons), where the adiabatic Pekar type polarons have a large radius. Anti-adiabatic (or non-adiabatic) polarons with a small radius have also been intensively investigated6. However, the anti-adiabatic limit is completely the opposite to the adiabatic limit discussed in our paper. In the adiabatic approximation, the adiabatic potential J, consisting of the electron energy E and the elastic energy of the lattice Eel = K P n Q2 n/2, is given by 2 J = Z d2x[ X n |∇ψn|2 2m⊥ + t X n |ψn −ψn+1|2 +D X n Qn|ψn|2 + K X n Q2 n 2 ] (2) The lowest energy self-trapped states correspond to a wave function which is homoge- neous in the transverse direction ψn(x, y) = ψn and is", + "= Z d2x[ X n |∇ψn|2 2m⊥ + t X n |ψn −ψn+1|2 +D X n Qn|ψn|2 + K X n Q2 n 2 ] (2) The lowest energy self-trapped states correspond to a wave function which is homoge- neous in the transverse direction ψn(x, y) = ψn and is a true solution of the above equation. Then the equations describing the self-trapped states are obtained by minimization of J with respect to ψn and Qn provided that P n R d2x|ψn|2 = 1 (see, for example,4,5). Doing so, ie by a minimization of J with respect to Qn, gives Qn = −D K |ψn|2, where K is the modulus of elasticity of the lattice. Substituting for Qn(ψn) in eq. (1), (2), respectively, gives the nonlinear Schr¨odinger equation (NSE) −ψn−1 + 2ψn −ψn+1 −c|ψn|2ψn = Eψn (3) and the corresponding expression for the adiabatic potential J = X n |ψn −ψn+1|2 −c 2 X n | ψn |4 (4) where c = D2/tK is the dimensionless electron-phonon coupling constant and E and J are measured in units of the miniband width t. The normalization condition is P n | ψn |2= 1/L2, where L2 is the area of the transverse plane of the SSL (assuming it is a square of side L). It is convenient to make the scaling transformation ψn →ψn/L. This transformation does not alter the form of the equations but it does give the conventional normalization condition for the new wave function P n | ψ2 n |= 1. However, the coupling constant does change so that cnew = cold/L2. In a longitudinal magnetic field the continuum transverse spectrum of the miniband is split into Landau levels. For strong magnetic fields only the first Landau level is relevant. The electron on this level is localized in a transverse direction in an area of the order of l2 B, where lB is a magnetic length, such that, l2 B = C¯h/eB (L and lB are measured in units of the SSL period d). Then the electron motion is only one dimensional along the direction of the magnetic field through the SSL. The coupling constant then transforms to cnew →cold/l2 B. Thus, with the increase of the longitudinal magnetic field, the phonon coupling constant can be strongly enhanced. To illustrate, consider some examples having exact solutions. The simplest is a double well structure. For a large coupling constant, c ≫1, there arises the trapping of the electron in one of the two wells. The eigenvalue of this state is E = 2 −c and the corresponding eigenvectors are: ψ1 = √1±α √ 2 and ψ2 = √1∓α √ 2 where α = q 1 −A c2, where A = 16 or A = 4 for periodic (PBC) or open boundary conditions, respectively. These states are two-fold degenerate. Another straight-forward example, studied analytically, consists of three superlattice layers or three quantum wells. The ground state is characterized by the localization of the electron in a single quantum well in the same manner as in the double well structure. For PBC with an", + "respectively. These states are two-fold degenerate. Another straight-forward example, studied analytically, consists of three superlattice layers or three quantum wells. The ground state is characterized by the localization of the electron in a single quantum well in the same manner as in the double well structure. For PBC with an increase in the value of c this state very rapidly saturates to the limit {ψ1, ψ2, ψ3} ⇒{0, 1, 0} or {0, 0, 1} or {1, 0, 0}. These three states are degenerate due to the translational symmetry associated with PBC and corresponds to the ground state energy 3 which in the limit c →∞takes the simple form E = 2 −c, ie the same as the double well structure. There is also an excited self-trapped state, which, for c ≫1, has the (degenerate) structure {ψ1, ψ2, ψ3} ⇒{1/ √ 2, 0, 1/ √ 2} or {1/ √ 2, 1/ √ 2, 0} or {0, 1/ √ 2, 1/ √ 2}. This state corresponds to the eigenvalue E = 1−c 2 and the region of this self-trapped localization is two neighboring wells. There is also another self-trapped state in this system, which arises for very large values of c. This state has the eigenvalue E = 3 −c 2 and the associated structure has the form {ψ1, ψ2, ψ3} ⇒{ 1 √ 2, −1 √ 2, 0} or analogous cyclic permutations of this pattern. An exact solution, valid for all values of c, has also been found, however, it does not appear to have a simple form. Consequently, the exact solutions are presented graphically in Fig.1. The self-trapping states emerge for values of the coupling constant, c > 3. From this figure it is clear that the solutions obtained in the limit c →∞describe the spectrum of the triple quantum well structure very accurately. The first corrections to the presented eigenvectors are of order 1/c. Without PBC the translational invariance is broken and the degeneracy is lifted so that the number of energy levels increases. The exact spectrum for the triple quantum well structure, without PBC, is presented in Fig. 2. These new levels may also be described analytically for c ≫1. A new eigenvalue, E = 2 −c 2 appears, which is associated with the localized self-trapped solution { 1 √ 2, 0, ±1 √ 2}. Additional eigenvalues, which also satisfy the NSE, include E = 2−c 3 , 6−c 3 , 10−c 3 . These eigenvalues are associated with states where the electron is localized in each of the quantum wells with equal probabilities. Again, for nearly all values of coupling constant c, the amazing coincidence between the spectra obtained for c ≫1 and the exact result shown in Fig.2 is seen. From the examples discussed above, it is noticed that in the limit c →∞, the spectrum has some special features associated with the pattern of the localized states. The spectrum depends on how large the set of localizing quantum wells is and how many spots (groups of neighboring, localizing quantum wells) this set is separated into. The lowest", + "that in the limit c →∞, the spectrum has some special features associated with the pattern of the localized states. The spectrum depends on how large the set of localizing quantum wells is and how many spots (groups of neighboring, localizing quantum wells) this set is separated into. The lowest eigenvalue always has the form E = 2 −c and is associated with the localization of the electron in a single quantum well. The next eigenvalue, E = (2−c) 2 , is associated with the localization of the electron in a single spot consisting of two neighboring quantum wells. However, when the localization occurs in two separate quantum wells, so that there are two spots each consisting of one well, the energy eigenvalue is E = (4−c) 2 . With the localization of the electron in three quantum wells the associated eigenvalue may be written as E = 2−c 3 . This observation can be used to derive exact solutions associated with an arbitrary localization pattern of an electron in SSL consisting of N quantum wells. Initially, in the zero approximation with parameter c ≫1, assume that the electron is localized in a single quantum well of a N−well SSL, say the kth well, so that |ψk|2 = 1 and |ψn|2 = 0 for all possible n ̸= k. Upon making this substitution into the NSE and after some algebraic manipulation, we see that this substitution corresponds to a solution with the eigenvalue E = 2−c. Similarly, if it is assumed that the electron is localized in two neighboring quantum wells, say, the kth and (k + 1)th, and we substitute into the NSE ψk = 1 √ 2, ψk+1 = 1 √ 2 and ψn = 0 for n ̸= k, n ̸= k + 1 then (after some algebra) the eigenvalue E = 2−c 2 is obtained. For the localization of the electron in 3 neighboring wells, we make the substitution ψk = ψk+1 = ψk+2 = 1 √ 3 and ψn = 0 for all other wells. This then yields E = 2−c 3 . Repeating the same procedure for the electron localized in n neighboring wells, we derive the eigenvalue E = 2−c n . 4 The same procedure may be applied for an electron localized in two separate spots, consisting of, say, n1 and n2 quantum wells. Surprisingly, we find that in this case, E = (4−c) n , where n = n1 + n2. If this localization occurs in m distinct spots, and each spot consists of n1, n2, . . . , nm neighboring quantum wells, the associated eigenvalue is E = (2m−c) n , where n = n1 + n2 + . . . + nm. However, this is still not a complete set of solutions. Note that if the electron is localized in a spot consisting of n1 = 2n0 neighboring quantum wells, then the wave function may change sign for the different quantum wells of this spot. For example, if for the first n0 neighboring quantum wells the wave function has a", + "of solutions. Note that if the electron is localized in a spot consisting of n1 = 2n0 neighboring quantum wells, then the wave function may change sign for the different quantum wells of this spot. For example, if for the first n0 neighboring quantum wells the wave function has a positive sign and for the remaining n0 wells the wave function is negative, then the associated eigenvalue is E = 2m+4−c n . In general, if the wave function changes sign l times in one or several localized spots the energy eigenvalue of the NSE takes the form E = 2m + 4l −c n (5) where n is the total number of quantum wells in which the electron is localized. Note that, in each localizing quantum well the electron is localized with equal probability, ψ2 k = 1/n for all numbers k associated with localizing quantum wells. This probability does not depend on the number of spots, m, the localization area is separated into. The number n may take any integer value n = 1, 2, 3, . . . , N. The number m may take any integer value satisfying m ≤n. Similarly, the number l satisfies l ≤n −1. Eq. (5), obtained in the limit c →∞, has been compared with the exact solutions for systems consisting of up to 5 quantum wells, in the same manner as for the triple quantum well structure. Also a comparison has been made between energy spectra obtained from equation (5) and corresponding eigenvectors and energy eigenvalues from numerical calcu- lations for SSL consisting of up to 11 quantum wells. In all these cases for nearly all values of c (except small regions of critical values where the self-trapped solutions originate) there is perfect agreement with the derived formula (5). However, in contrast with this perfect agreement between eigenvalues, a decrease in the value of c leads to a noticable deviation in the wave functions (eigenvectors) from those obtained in the limit c →∞. The first order corrections to the presented eigenvectors obtained with the use of perturbation theory is of the order of √n/c. When the coupling constant c is not very large the wave functions of some states may have interesting incommensurate and chaotic structures. The detailed analysis of such wave functions will be published elsewhere7. Thus, from the comparison with numerical results and with perturbation theory we conclude, that even though the spec- trum of the NSE for the system with a finite, arbitrary number of wells N is well described by equation (5) the shape of appropriate wave functions for smaller values of c may have only qualitative features of the appropriate eigenvectors obtained in the limit c →∞. As c decreases the localization spots smear out and the boundaries between localizing quantum wells and quantum wells where the wave function vanishes diminish. In general, for large ranges of the values of c the eigenvalue Elmn corresponds to the state of an electron which is self-trapped or localized with near equal probability in n wells which are separated into", + "boundaries between localizing quantum wells and quantum wells where the wave function vanishes diminish. In general, for large ranges of the values of c the eigenvalue Elmn corresponds to the state of an electron which is self-trapped or localized with near equal probability in n wells which are separated into m groups (spots). Between the spots the electron wave function is nearly vanishing while inside these spots the wave function changes sign l times, although its amplitude is nearly the same. Since the spectrum, eq.(5), is associated with a local localization pattern, it is universal and does not depend on the boundary conditions. 5 The eigenvalues Enml are electron energies created in SSL by local deformations. There- fore, it is interesting to estimate an adiabatic potential, Jnml = Enml + Eel, needed to create these states. With the use of the method described above the following expression for the adiabatic potential is obtained, Jnml = 4m + 8l −c 2n (6) For comparison, a state associated with the bottom of the miniband shifted by electron- phonon interaction corresponds to an adiabatic potential Jbs = −c/(2N). For a single localization spot with no change in the sign of the wave function, l = 0 and m = 1 are substituted into (6). This corresponds to the lowest value of adiabatic potential for all values of n. From these equations one sees that for strong coupling, c ≫1, there are always self-trapped states which have a lower adiabatic potential than analogous miniband states, ie J110 < Jbs. However, the states with very wide spots may have Jn10 > Jbs. This will occur when the size of the spot, n, becomes larger than some critical value nc = N(c −4)/c. The presence of these states (with Jn10 > Jbs) indicates that the self-trapped states associated with the quantum numbers n < nc and the band states are separated by a self-trapped barrier, which appears only because of the lattice discreteness and finite size of the system. The self-trapping arises for a critical coupling equal to ccrit = 4N/(N −1). In the continuum 1D case the self-trapped barrier is absent although the critical coupling for the self-trapping does exist and depends on the size of the system8. The self-trapped barrier exists only in the continuum 3D case, as was first discovered by Rashba2,4. Thus, we have described a new phenomenon of the self-organized, deformational creation of single, double, triple and in general, multiple quantum well structures in SSL with an electron locked inside. This is a local effect which depends on how many wells the SSL consists of (but exists for any number of wells) and on how perfect the SSL is. It may, therefore, be readily observed in experiments. The states with the spectrum as described by equation (5) may be detected, for example, by resonant tunneling experiments or by absorption of light. Each self-trapped state associated with different quantum numbers n, m, l will give rise to an absorption peak which lies below the miniband and is well described by equation (5). However, as", + "described by equation (5) may be detected, for example, by resonant tunneling experiments or by absorption of light. Each self-trapped state associated with different quantum numbers n, m, l will give rise to an absorption peak which lies below the miniband and is well described by equation (5). However, as the self-trapped states are associated with deformations, which maintain the localization, the main absorption of light (which follows the Franck-Condon principle) will be associated with the miniband. Since the creation of the self-trapped states takes less energy than the creation of the band states (see equation (6)), after absorbing radiation, the electrons excited at the absorption of light inside the miniband then may be self-trapped. The emission of radiation from these states will directly indicate their presence. The creation of excitons from these states is an interesting issue. Another interesting feature of the self-trapped (or locked) states in SSL is their strong interaction with phonons or with ordinary sound waves. Each of the localizing spots of a self-trapped state is created by deformation and therefore also associated with the phonon localisation. Since inside the spot the deformation ∼1/n and vanishes outside then each spot may be viewed as a sound wave resonator. The resonant frequency and the wavelength of this resonator is related to the size of the spot. For example, for the spot consisting of ni quantum wells the size is nid (note that d is a period of SSL and P i ni = n). Then the resonance wavelength will be λ = nid and the phonons with the wavevector 2π/(nid) will be 6 especially strongly scattered or trapped by the localizing spot. Based on this fact one may suggest the following scenario for experiments. When a sound wave moving through a SSL passes the self-trapped spot there will be a transfer of momentum from the sound wave to the spot. As a result of this scattering, the acoustic phonons with wave vector q = 2π/(nid) will be absorbed forcing the self-trapped spot to propagate through the lattice. Such motion may give rise to the creation of current induced by sound. Alternatively, if a bias voltage is applied to the SSL, creating a current, a constant backflow of appropriate phonons will be created. This backflow of selective phonons will be especially strong at some bias voltages equal to the interlevel spacing. Therefore, they may be observed as some minima in the current voltage characteristics of the SSL in a high magnetic field. This effect may exist only at low temperatures when there are no current carriers in the miniband. Otherwise, in doped SSL or at high temperatures the transport will be of the Bloch type. Since the number of self-trapped states increases with the total number of quantum wells in the SSL it is expected that the proposed effect will be stronger in SSL with larger numbers of quantum wells. Since the described effect of the locking of an electron in quantum wells is caused by acoustical phonons it is universal and should not depend on what material the SSL", + "the SSL it is expected that the proposed effect will be stronger in SSL with larger numbers of quantum wells. Since the described effect of the locking of an electron in quantum wells is caused by acoustical phonons it is universal and should not depend on what material the SSL consists of. Recently we have become aware that a similar effect has been seen by L. Eaves9 group. In summary we have described novel, self-trapped states of an electron associated with the self-organized, deformational creation of single, double and multiple quantum wells which lock the electron in SSL. The exact energy spectrum associated with these states for a strong electron-phonon interaction, D2/(Kt) ≫1, or for a strong longitudinal magnetic field is derived. The novel states associated with local deformations have very different properties from the conventional band states and therefore may give rise to some new effects like, for example, the creation of an electric current by sound. Alternatively, such states stimulate sound absorption and prevent phonon propagation. This may give rise to unusual thermopower in SSL. 7 REFERENCES 1 S.I. Pekar, Untersuchungen ¨uber die Electronentheorie des Kristallen, Akedemie Verlag, Berlin, 1954. 2 E.I. Rashba, Opt. Spectr. 2, 78, 2, 88 (1957) 3 T. Holstein, Ann. Phys. 8, 343 (1959) 4 E.I. Rashba, in: Excitons, ed. by E.I.Rashba and M.D. Sturge, North-Holland (Amsterdam) 1982, p.543. 5 F.V.Kusmartsev, Phys.Rev. B43, (1991), 1345. F.V.Kusmartsev et al, Europhys. Lett. 42 547 (1998) 6 A.S.Alexandrov and N.F.Mott, Polarons and Bipolarons (WS, Singapore) 1995 7 H.S. Dhillon, F. V. Kusmartsev, and K. E. K¨urten, in preparation and cf, also, in F. V. Kusmartsev and K. E. K¨urten, Effects of Chaos in Quantum Lattice Systems in Lecture Notes in Physics, Vol. 284 (1997), edited by J. W. Clark and M. L. Ristig, Springer-Verlag, NY-Heidelberg Berlin 8 E.I. Rashba, Synth. Met. 64, 255 (1994) 9 R. Hayden, L. Eaves et al,(private communications) September 1998 Figure Caption Fig. 1 The dependence of energy spectrum on the electron-phonon coupling constant c for a triple quantum well structure obtained assuming PBC. The lines of the spectrum correspond to (from bottom up on left hand side) the eigenvalues E = 2−c, E = (2−c)/2, E = (6−c)/2, E = −c/3 and E = (8 −c)/3. Fig. 2 The same as in Fig.1 for a triple quantum well structure obtained without assumption of PBC. The lines of the spectrum correspond to (from bottom up on LHS) the eigenvalues E = 2 −c, E = (2 −c)/2, E = (4 −c)/2, E = (6 −c)/2, E = (2 −c)/3, E = (6 −c)/3 and E = (10 −c)/3. 8 -10 -8 -6 -4 -2 0 Energy Eigenvalue 0 5 10 15 20 25 30 35 Coupling Constant Fig 1: Spectrum For 3 Quantum Wells Untitled-2 1 -10 -8 -6 -4 -2 0 Energy Eigenvalue 0 10 20 30 40 Coupling Constant Fig 2: Spectrum For 3 Quantum Wells Untitled-1 1", + "1 -10 -8 -6 -4 -2 0 Energy Eigenvalue 0 10 20 30 40 Coupling Constant Fig 2: Spectrum For 3 Quantum Wells Untitled-1 1", + "arXiv:1712.01630v1 [physics.acc-ph] 5 Dec 2017 Subrelativistic electron source for Dielectric Laser Acceleration, elements of design Jean-Luc BABIGEON∗ Contents 1 Summary 3 2 Dielectric Laser Acceleration for the future projects of high energy linacs 3 3 Analysis of available electron emission techniques 4 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Field emission induced by electrical voltages . . . . . . . . . . . . 4 3.3 Field emission induced by laser . . . . . . . . . . . . . . . . . . . 5 3.4 Our project : Field assisted photo-emission in pulsed configuration 6 3.4.1 Principal features . . . . . . . . . . . . . . . . . . . . . . . 6 3.4.2 Consequences on the phase choice of laser . . . . . . . . . 8 4 Components of sub-relativistic electron source 9 4.1 Pulsed electrical source . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Adaptation of input cathode impedance . . . . . . . . . . . . . . 11 4.3 Nano-structured photo-cathodes and Field Emitter Arrays (FEA) 13 4.4 nano displacements and spatial alignments between components . 15 4.5 Electron optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.6 DLA first stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.6.1 THz cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.6.2 Traveling wave and direct acceleration in near infrared laser range . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.7 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.7.1 Charge of a bunch . . . . . . . . . . . . . . . . . . . . . . 25 4.7.2 Bunch length . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.7.3 bunch position and lateral dimensions . . . . . . . . . . . 27 4.8 Summary and discussion on possible and preferred FEA and DLA configurations . . . . .", + ". . . . . . . . . . . . . . 27 4.7.3 bunch position and lateral dimensions . . . . . . . . . . . 27 4.8 Summary and discussion on possible and preferred FEA and DLA configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Conclusion 30 ∗Laboratoire de l’Accélérateur Linéaire - 91898 - Orsay, France; babigeon@lal.in2p3.fr 1 List of Figures 1 thermal and field emission regimes, and transition zones, after [1] 5 2 Optical pulse combined with HV pulse . . . . . . . . . . . . . . . 8 3 Optical pulse combined with RF pulsed amplifier . . . . . . . . . 8 4 DC-pulsed principle . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Layout of primary current switch . . . . . . . . . . . . . . . . . . 10 6 Direct current and pulse timing of DSRD generator, from [2] . . 11 7 Equivalent circuit of the coaxial gun . . . . . . . . . . . . . . . . 12 8 Input impedance of the coaxial gun . . . . . . . . . . . . . . . . . 12 9 Spindt-like FEA 1D Cathode . . . . . . . . . . . . . . . . . . . . 14 10 Carbon Nano-tube FEA 1D cathode . . . . . . . . . . . . . . . . 14 11 geometry and freedom degrees in case of independent components inside gun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 12 Efield transient tip Response to laser gaussian 100fs pulse, oppo- site plane, times 95 (up) and 139fs . . . . . . . . . . . . . . . . . 20 13 Coaxial gun version . . . . . . . . . . . . . . . . . . . . . . . . . 21 14 Integrated version . . . . . . . . . . . . . . . . . . . . . . . . . . 21 15 General case of Dielectric Laser Acceleration . . . . . . . . . . . 24 16 miniature Faraday cup . . . . . . . . . . . . . . . . . . . . . . . . 26 17 Standard device (from Kindall) . . . . . . . .", + ". . . . . . . 24 16 miniature Faraday cup . . . . . . . . . . . . . . . . . . . . . . . . 26 17 Standard device (from Kindall) . . . . . . . . . . . . . . . . . . . 26 18 Setup with THz cell . . . . . . . . . . . . . . . . . . . . . . . . . 28 19 Coaxial setup with DLA . . . . . . . . . . . . . . . . . . . . . . . 28 20 Integrated setup in strip line configuration . . . . . . . . . . . . . 28 2 1 Summary We describe and discuss solutions for an innovating sub-relativistic electron source, based on pulsed DC gun, including linear (1D) Field Emitter Array (FEA) associated immediately with Laser Dielectric Accelerator (DLA) stage. That setup is designed to avoid any standard accelerator component (electro- static or magnetic macroscopic dipole . . . ) Parts of the diagnostics, including spectrometer and all diagnostics, are planned to rely on nano-structures. Then the entire setup -pulsed source, laser and measurements ends excepted- should be enclosed inside a decimeter range vacuum chamber. The goal of that paper is to suggest experimental orientations, which are discussed inside our laboratory. However we shall also notice the theoretical issues among which we actually work, these one could be presented in a next work. 2 Dielectric Laser Acceleration for the future projects of high energy linacs Laser driven acceleration in Dielectric (DLA) principle is not a new concept, it has been described -at less- soon as 1996 for instance by [3]. The “Accelerator- in-chip”, under impulse of Pr Byer, has been schematically described by [4]; assuming the occurence of thousand of low cost lasers and dielectric amplifiers, each amplifier of 1mm characteristic length, it could be possible, for example, to reach the TeV range, independently of numerous other industrial and research applications. This concept intends to solve following practical considerations or questions : 1. is it possible to stop inflating infrastructure sizes/costs for TeV electron accelerators ? 2. Can we exploit the emerging market of laser sources for establishing new concepts and performances in accelerators ? 100TeV proton collider made by standard technology could amount up to 190km infrastructures with copper cavities, klystrons, standard magnetics... And added cumbersome cold technologies [5]. The goal of alternative tech- niques is to reduced the infrastructures, at less for 1TeV electrons colliders, to much less than km range Linac. In this contribution, we describe the very first stage 0-10keV, of a 1TeV Accelerator-in-chip, ie sub-relativistic electron source in the 10keV range. In- deed, it is considered to be the most difficult step, considering the spatio- temporal exploding character of electron bunch at low energies, particularly at the demanding total charge for that type of source. It is probably why first", + "a 1TeV Accelerator-in-chip, ie sub-relativistic electron source in the 10keV range. In- deed, it is considered to be the most difficult step, considering the spatio- temporal exploding character of electron bunch at low energies, particularly at the demanding total charge for that type of source. It is probably why first efforts were directed on relativistic dielectric stages, inserted inside conventional accelerators [6][7][8]. From other side, if we demonstrate that in contrary, we are able to reach low emittance figures at low energy level, it will considerably help for instance, the implementation of low cost free electron lasers [9]. Beyond emittance, the accelerator figure of merit is -considered to be- lu- minosity so we are to combine emittance with enough charge. In a first intro- ductory section 3, we describe the principle of our femtosecond electron source, 3 and specify the desirable performances, relying on field assisted photo-emission induced on Field Emitter Arrays (FEA). In the following section 4, we analyze successively the components of our source, pulsed photo-cathode, photonics, dielectric accelerator, and finally mea- surements; we point and separate numerous technical and theoretical problems associated, considering that it should not be pertinent to develop specific theo- retical point if the design is not prealably defined at less in their broad lines. Finally in section 5, we sketch a working schedule, and discuss about the ressources and efforts to devote to it. 3 Analysis of available electron emission tech- niques 3.1 Overview They are several possible regimes of emission in vacuum, cold field (tunneling), Schottky (thermally and field enhanced emission) and strong Field Emission (FE) (tunneling), depending on physical mechanisms. Instead plotting standard potential description like [10] (fig 2 of his lecture), we emphasize by figure 1 from [1], the intricate role of thermal component, which acts like a transition zone between pure Thermo-emission and Field emission. The n coefficient (see reference), n = βT βF is a marker of the predominance of each regime. n >> 1 is the field (F) emission domain, and n << 1 the thermal (kT) one. This interesting Jensen’s idea may participate in the basis of FEA design, concurrently with abinitio considerations. In the general case, we extract the electron by electromagnetic potential difference, induced either by electrical voltage, or by laser photo-emission. In the following, we discuss the applicability of different regimes to our specific source. 3.2 Field emission induced by electrical voltages The strong FE regime is assumed to occur above a near field threshold in the range 3 to 10GV/m near FEA tips [11]. In that strong field mode, some “ther- mal” effects occur. The acceleration field in strong FE regime, is also prone to generate high charge, high energy anomalous electron in the front of the bunch,typically 20% higher than normal arrival energy, in the case of Diode guns [12]. With Diode guns, that outcome is interesting if we accept low repetition rates, with charges much larger than 1nC/bunch, given the time for the cathode to recover its original state, if it is yet possible, excluding electro-chemical induced effects. Although that type of", + "in the case of Diode guns [12]. With Diode guns, that outcome is interesting if we accept low repetition rates, with charges much larger than 1nC/bunch, given the time for the cathode to recover its original state, if it is yet possible, excluding electro-chemical induced effects. Although that type of source avoid the cost and use of a laser, it is rather oriented to high charge accelerators, like induction one, with nanosecond bunches. Some interesting experimental efforts -to follow if technology evolutes...- were devoted to production of sub-nanosecond electron pulses, with as high as several hundred of keV bunches [13]. Anyway, the cathode recovering under such pulsed Ultra High Voltages seems to exclude more than 10Hz repetition rate, with limited cathode time life. We convinced ourself that it is also not trivial to generate picosecond emission with high PrF only by electrical pulses, so the choice is 4 Figure 1: thermal and field emission regimes, and transition zones, after [1] today oriented with laser assistance for ultra-fast bunches in accelerator -and spectroscopy- communities. The inception of FE occurs in a tiny interval of voltages, due to non linear laws of emission. Under that threshold lies also Schottky mode. In that last mode, a few of electrons are emitted, either directly by the cathode or by the corrugations over the metallic walls of the gun, their distribution forming the “black” current. Much electron sources for accelerators live with Schottky emis- sion, RF standard guns included. However, the presence of FEA is expected to enhance cold FE against dark current. Moreover, regarding dielectric accelera- tors, the tiny dimensions of electron sources is one reason to choose a voltage range between 10 and 30kV partly because of breakdown consideration and/or thermal degradation. The voltage, in fact is obviously not the real marker, rather the emitted current. So it is admitted to use MV pulses, but with cm range cathode-anode distances, at the cost of low PrF. For instance, in [14], a distance apex-anode of 1cm for 50kV, with apex of 1µm and tip length of 1cm, was linked with fields of 5109V/m. 3.3 Field emission induced by laser Regarding photo-emission, many terms are also often used, field assisted, Schot- tky assisted, Photo-field photo-emission, above threshold ionisation, optical tun- neling. . . . The combination of electrical (HV) work and laser energy overcomes the work function barrier and permits electron emission in vacuum. Let’s recall as in [11], that the major criterion to distinguish strong FE 5 induced by laser, from multi-photon like Schottky is the Keldysh factor. To summarize, electronic emission may be generated 1, by pulsed electrical source (diode guns), by HV source assisting photo-emission (DC guns or photo- field emission if electrical field is not present), or by photo-emission synchronized with electrical HV pulse (pulsed DC guns). 3.4 Our project : Field assisted photo-emission in pulsed configuration 3.4.1 Principal features That present contribution is oriented to field assisted photo-emission by a elec- trical pulsed-DC gun. The choice of pulsed voltage is suggested by the techno- logical performances of high PrFs as precised above.", + "pulse (pulsed DC guns). 3.4 Our project : Field assisted photo-emission in pulsed configuration 3.4.1 Principal features That present contribution is oriented to field assisted photo-emission by a elec- trical pulsed-DC gun. The choice of pulsed voltage is suggested by the techno- logical performances of high PrFs as precised above. 2 We have followed the same original principle than [15] but with following modifications : 1. Laser is femtosecond instead of picosecond one, 2. Nanosecond voltage is 10 kV (≫MHz) instead at 60kV(30Hz), 3. Voltage waveform is bipolar, 4. Emitter is 1D CNT FEA instead of single ZrC tip, with individual apexes of 20 to 50 nm instead of 1µm, but length of the tips may reach 1µm, 5. There are two possible designs for the gun, coaxial one like [15] and inte- grated with DLA 3 Despite its -at first glance- added complexity, the pulsed setup/mode offers several advantages : 1. the breakdown level is higher, as it is well explained by [16], who defines a specific criterion for transient AC fields, with levels beyond Kilpatrick threshold, 2. the combination of pulsed laser and (fast) pulsed voltage allows to isolate and cut the electron bunch to ultra short longitudinal emittances before the acceptance of DLA. Indeed, with the first DC guns setups used for demonstrator of DLA acceleration [6], it was shown that only a low frac- tion of the beam -emitted permanently in case of DC guns-, was in the proper phase for acceleration. While it was convenient for experimental evaluation, extrapolation to real source should ask for other solutions. 1depending on amplitude, duration, source impedance, power . . . of electrical or optical sources; for instance, with ultra short 1MV UHV electrical pulse, we may meet tunneling field emission type, while with 40kV 1µs electrical pulse with high current, Schottky type may occur, etc. 2Some features are common with ultra-fast spectroscopy experiments 3in the following, we nevertheless shall denote the “field emission”, the global effect of the two fields, emphasizing that total electric field sums laser induced and electromagnetic one, induced by pulsed electrical mode 6 3. with a bipolar electrical pulse, we probably reduce the ionic bombardment on the cathode, because the mean value of accelerating forces on ions is null inside a bipolar cycle; furthermore, it could help to discharging the dielectric surfaces of the first accelerating stage, and so enhance the efficiency of acceleration, 4. one can easily show [10] that Schottky effect induced by pulsed source reduces the necessary gap for electrons to be emitted in vacuum 4; either one can see it as improvement of quantum yield of photo-emission, for a given laser wavelength, either as a technique for working with the same quantum yield, but in the near infrared range, instead of costly near UV range, 5. fast pulsed cathode mode is anticipated to lighten other physical mecha- nisms that electrostatic field of the DC gun; we shall come back on it, in section 4, precisely inside design of photo-cathode. The superposition of electrical and optical pulses is shown in figure 2 and", + "near UV range, 5. fast pulsed cathode mode is anticipated to lighten other physical mecha- nisms that electrostatic field of the DC gun; we shall come back on it, in section 4, precisely inside design of photo-cathode. The superposition of electrical and optical pulses is shown in figure 2 and 3 in the two cases of HV pulsed and RF pulsed sources. Of course, technol- ogy may enlarge the limits, given by thermal and power considerations. We represent the expected waveform diffracted by FEA, not the initial laser Wave- form, which could be for instance a Gaussian one. Indeed, it is very probable that any unipolar waveform is not conserved by interaction with high pass ele- ments like monopoles/dipoles of the FEA. Note that these representations are for demonstrative goal only. In HV figures, range is ns, and in RF one, µs. In our study of DSRD pulsed generator, we had not precised the exact width of the “flat top”. In fact, with a single stage, we expect a width of 10ns, with transition times 0.5ns approximately. With a second compression stage, we estimate the waveform to be finally 10kV/1ns/200ps, these figures explaining by themselves. So in the figure hereunder, we took in hypothesis that last performance. The PrF of the HV generator is taken to be 1MHz. Laser oscillator available at Lal has 80MHz PrF, ie 13ns inter-pulse period, so we have to divert 1/80 of laser impulses. For the RF pulsed amplifier, we have borrowed the model of 400W S-band amplifier for klystron, with 2 microsecond width. In that case, we can inject roughly 10 laser pulses inside one RF pulse. Of course, these figures are not to be definitive, for instance, an X-band amplifier is conceivable and power is chosen by availability criterium. Our RF amplifier design capabilities are given by the transistor one, ie 100µs/400W/20%, which result in 2kHz PrF. We notice that only 2µs were specified, not 100, the reason why being our care about possible thermal effect inside amplifier. If we keep anyway, the PrF, we can extrapolate the performances of each present power sources, regarding the number of pulse/s, and the Lal capabilities. 1. in HV case : 1Mhz with 1/80 laser pulses, in multi-burst mode, 20 pulses by burst with 100Hz; in the hypothesis on dividing laser frequency, we may expect 20x100 = 2000 pulses/second, 2. in RF case : 10 laser pulses inside 1 RF one, Prf 2kHz, so 10x2000 = 20000 pulses/second. 4but we admit with that author, that we cannot reduce energy gap to zero! 7 Figure 2: Optical pulse combined with HV pulse Figure 3: Optical pulse combined with RF pulsed amplifier * : One may ask if space charge, considered as minor inside a single pulse, could not occur with high PrFs. But the principal issue today felt is the possi- bility of induced electromagnetic effects from inter pulses or bunches-structures interactions. The figure 4 describe schematically the pulsed-DC gun technique. 3.4.2 Consequences on the phase choice of laser As the laser pulse is superimposed on electrical one, guessed", + "with high PrFs. But the principal issue today felt is the possi- bility of induced electromagnetic effects from inter pulses or bunches-structures interactions. The figure 4 describe schematically the pulsed-DC gun technique. 3.4.2 Consequences on the phase choice of laser As the laser pulse is superimposed on electrical one, guessed perfect synchro- nism, it is known that efficient temporal compression of the bunch occurs de- pending of relative phase of laser injection. The Forbes non linearity of Schottky emission could help that compression. However, in HV case, it becomes then impossible to work with multi-pulse concept, placing several laser pulses on “flat top” of nanosecond electrical pulse, because evidently all pulses become 8 Figure 4: DC-pulsed principle different. Moreover, there is a supplementary possibility to explore with tuning the phase, ie to study the Schottky effect occurrence linked with 1 or multi photon extraction, and very low emittance [17], knowing we opt for a non energetic laser regime, where thermal effects are to be minimized and Keldish factor high. All these considerations are known in electron source domain. However they are to be faced to new experimental design. 4 Components of sub-relativistic electron source 4.1 Pulsed electrical source Pulsed source is a High Power Pulsed (HPP) High Voltage (HV) generator, with sub nanosecond rise and fall times. In its principle, it follows the DSRD (Drift Step Recovery Diode) generator made by [2]. We have studied such a generator at Lal [18], the objectives being 100ps rise time at PrF higher than 1 MHz and for peak voltages of 10kV (which means peak to peak voltage of 20kV). The primary current PCB layout of the generator with 4 Mosfet switches in parallel is represented on figure 5. We have also analyzed the reduction of the jitter. With conventional fast electronic techniques, it seems possible to reach a < 5ps jitter. A lower one could be achieved, for instance in subps range, at the expense of rigorous voltage regulation, which is explained below. Indeed, the control of direct pulse current is fundamental for the DSRD diode. We show at figure 6, thanks to [2], the timing of pulse generation. * : Let’s come back to timing diagram of the pulse, in the figure 6. We can see qualitatively that the jitter will depend of the position of Primary current. In fact, if the current waveform is perfectly reproducible, the precise instant of current reversal is determined, and the diode response being somewhat deter- 9 Figure 5: Layout of primary current switch ministic compared to times scales, we can admit that jitter will be minimal, of course implicitly accepting a precise trigger. So the timing question depends principally of precision of current injection, and specifically of precision of maximum level, or maximum level of voltage charge. The emergence of new SiC HV diodes [19] brought us to investigate the semi-conductor physics to determinate if their use in DSRD generator is possible. Indeed, SiC diodes are presented like soft recovery rectifiers, for power market; it is the opposite quality we need, ie a component able to", + "charge. The emergence of new SiC HV diodes [19] brought us to investigate the semi-conductor physics to determinate if their use in DSRD generator is possible. Indeed, SiC diodes are presented like soft recovery rectifiers, for power market; it is the opposite quality we need, ie a component able to store energy from Direct current and give it back in a kind of “compression mode” during the opening phase. That concept of opening switch associated with inductive generator is a motivating technology. The dynamic behavior of the switch depends mainly of metallurgical profile of doping. A lot of convincing hypotheses bring us today to estimate with Grekhov arguments [20] that the P doped PIN diodes, P+pN are better fitted than P+nN one for DSRD effect. It has to be scientifically ascertained by futures simulations, physics models and specific SiC prototypes. Nevertheless, an other issue is the repetition frequency. Despite its tremen- dous value of 1MHz for DSRD HV generators, higher than any PrF of conven- tional capacitor discharge HPP HV pulse generators the requirement of acceler- ator community for luminosity could ask for 100MHz to GHz range of PrF (see section 3.4; it is a serious challenge. Some recent analysis drives us to consider the use of power RF amplifier. Let’s, in first, look at necessary power : in case of 50Ωadaptation, the instan- taneous necessary power is P = V 2 2 Z = 10kV 2 4∗50 = 500kW. Of course, in case of CW amplifier, instantaneous power is of same order than mean power. If 10 Figure 6: Direct current and pulse timing of DSRD generator, from [2] we wish to build it in L or Sband for instance, it is far from economical today possibilities, and not coherent with “accelerator-in-chip” philosophy. Note that the advantage is here to pulse generator, considering instantaneous power. Nevertheless, two factors may reduce the minimum power : 1. the laser PrF are rather in 100MHz range for the moment, 2. the input impedance of our gun is much higher than 50Ω Independently from collider specifications, the consequences of PrF reduction is that power will scale with PrF, so for typical 80MHz laser oscillator (Element 100 Ti:Sa from Spectra-Physics), the ratio is P80MHz P3HGz = 80MHz 3000MHz = 0.266 ie the necessary (here mean) power will be 133kW, which remains very high. We have now to evaluate the real input impedance of our cathode. 4.2 Adaptation of input cathode impedance The geometrical and electrical models of our field emission cathode are repre- sented on figures 7 and 8 Here Cgap is cathode-anode capacity, assimilating it to a plate condenser, ie Cgap ∼8.8510−2pF with the typical dimensions of our gun. If we see roughly an emission tip as a cylinder, the DC resistance is Re ≈0.35mΩ. We calculate also the Forbes non linear resistance for vacuum emission of that tip. 5. With that circuit model and the associated equivalent input admittance we computed the time response of current density, under application of electrical bipolar pulse. We represent in the figure 8, the equivalent input impedance, including", + "We calculate also the Forbes non linear resistance for vacuum emission of that tip. 5. With that circuit model and the associated equivalent input admittance we computed the time response of current density, under application of electrical bipolar pulse. We represent in the figure 8, the equivalent input impedance, including the short vacuum transmission line inside our loadlock, and an arbitrary length of transmission line between generator and loadlock. 5In vacuum,the corresponding wavelength to 100ps rise times is 3cm. It is the characteristic dimension of our electron gun. So, using the quasi-static approximation we are to stay aware of that 11 Figure 7: Equivalent circuit of the coaxial gun 10 8 10 9 10 10 5.0x10 8 5.0x10 9 10 2 1.2x10 2 |Z| Module de l’impedance ramenee 10 8 10 9 10 10 5.0x10 8 5.0x10 9 10 2 1.2x10 2 Zr Partie reelle de l’impedance ramenee 10 8 10 9 10 10 5.0x10 8 5.0x10 9 0 −0.4 −0.2 Zi Partie imaginaire de l’impedance ramenee Figure 8: Input impedance of the coaxial gun 12 The figure 8 represents real and imaginary parts of that input impedance, viewed from the entrance of the coaxial injection line, centered around 300MHz frequency of the signal spectrum, in the case of HV pulsed generator. Note that as wavelength is 3cm, it is expected that input impedance should pass by minima and maxima. The idea is to stay at a high level of returned impedance, say 1kΩfor instance. The advantage is that generator current is minimized. In other side, the allowed current to the tip is to be regulated, or limited by the source, in order to avoid thermal runaway and to improve reproducibility of emission. Of course, the precise impedance is not specified here, because the final design includes many tip emitters 6, but it is a qualitative and demonstrative point. An other conclusion of our approach, is that whatever the injection mode, by pulse HV generator or RF pulsed amplifier, the adaptation condition is manda- tory, and it implies a injection length line well defined, at less cm precision. Of course, it is always possible to add a last tuning element, for instance a cage capacitor. Last but not least, that adaptation should be better optimized. * : Adaptation is not of course a single figure, because at less, the Forbes impedance is level dependent during the bipolar pulse. Then, the determination of precise adaptation is not a trivial question, but a synthesis problem in linear or non linear scope. As it is well known, even the linear solution is not univoque, depending on transfer function choice. Among candidates, could be elliptical Jacobi transfer function because they assume the steepest frequency roll-off, so dispersion inside crystal is minimum. With 1kΩimpedance instead 50Ω, the ratio being 20, the necessary power of our source, evaluated at precedent section, could be 133kW 20 = 6.65kW. With 20% rate of the signal, we only need 1.33kW. To summarize, Pulsed HPP HV generator versus RF power amplifier presents some -known- advantages and drawbacks 1. For medium and high", + "ratio being 20, the necessary power of our source, evaluated at precedent section, could be 133kW 20 = 6.65kW. With 20% rate of the signal, we only need 1.33kW. To summarize, Pulsed HPP HV generator versus RF power amplifier presents some -known- advantages and drawbacks 1. For medium and high instantaneous powers above, say some kW, pulsed HV, or High Power Pulsed generator have undisputed yields, probably 90% against 60% for standard RF class, and they can reach 100kV range and more,[21], 2. For PrF above 100MHz, difficulties will arise for implementations of FPP generators, 3. synchronization and jitter remains a common issue for both solutions, 4.3 Nano-structured photo-cathodes and Field Emitter Ar- rays (FEA) Since the first nano-structured cathodes by Spindt, much pioneering work has been undertaken for instance by Dr Tsujino and its team for now twenty years [22]. They tested the diode and triode configurations. In triode one, a gate al- lows us to trigger a fast pulse, avoiding laser photo-emission. The performance of such triodes are today limited to picosecond bunches. The diode configu- ration, where the bunch is effectively emitted by femtosecond laser, has been 6inside our approximate study, it is not so difficult to incorporate N emitters in parallel, admitting the mutual distance between them sufficient to neglect mutual impedances 13 Figure 9: Spindt-like FEA 1D Cathode Figure 10: Carbon Nano-tube FEA 1D cath- ode also studied by Toulouse with a DC gun [11]-generally made with a microscope 100kV basis- and by Hommelhoffteam. In that section, we focus on cathode technology, ie the materials and structure. Two configurations are showed in figures 9 and 10 : a spindt-like one, and a new proposed one, relying on Carbon family. Spindt-like cathode is studied far from now [23], [24] and constitutes a good reference comparison to any new candidate. * : Since we have insinuated that some additional physics must be con- sidered in sub nanosecond pulsed-DC guns, we shall only mention here, non thermal/cooling Nottingham effect[25] and plasmonic field enhancement, at the tips level. However, before describing the spatial domain around the tips, at the immediate vicinity of vacuum, it is fundamental to go “inside” the -photo- cathode and describe its physics with help of abinitio simulations. Indeed, phonons and surface state contributions for instance, strongly modify moment and energy distribution at that immediate interface. The new photo-cathode, based on carbon family, differs from initial Tung- stene, Molybdene, . . . Spindt FEA by many aspects : 1. a thin layer of graphene in Bernal configuration, 2. growing on graphene, of regularly spaced CNT in a 1D (linear) FEA * : as it is intuitive that a 1D linear array will perform optimally in emit- tance, in one direction X and badly in Y, the quantitative gain has to be better explained. It is expected that graphene constitute an electron reservoir for the CNT. The transition between conductive cathode and CNT needs to have some electri- cal resistance, in order to limit current in CNT. Incidentally, graphene layer pro- tects against eventual back ionic collision on the", + "has to be better explained. It is expected that graphene constitute an electron reservoir for the CNT. The transition between conductive cathode and CNT needs to have some electri- cal resistance, in order to limit current in CNT. Incidentally, graphene layer pro- tects against eventual back ionic collision on the cathode. CNT are proposed be- cause of their outstanding thermal conductivity, and their ratio height:diameter, raising more than 1000 [26] Several questions arises against that proposal, among them : 1. considering copper substrate coming from our gun, how will be the com- patibility between copper and graphene (see some answer elements in [27]) ? 14 2. how to grow a regular array of CNT on graphene (see some answer ele- ments in [28]) ? 3. how are the characteristics of graphene and CNT for efficient emission ? 4. can we ascertain these characteristics in fabrication process ? * : although these questions may be appreciated mainly as fabrication spe- cific, they have their counterpoint in theoretical and computational sides. In- deed, for instance, the compatibility copper-graphene is a wave-function prob- lem, and is studied with abinitio tools [29]. The CNT performances also, ie thermal capacity and 1D transport features are also belonging to solid state physics and described by the same tools [30]. * : One issue is the representation of ultrafast transients with the wave func- tion using TDDFT [31]. Last but not least, excited states and there hamiltonian representation -if correct-, are a theoretical today topic. Let’s describe qualitatively the expected behavior of the assembling copper- graphene-CNT under high electric fields. In static simple description, there is a voltage induced between each nanotip and the graphene layer in its immediate proximity. It is furthermore guessed that a voltage will be present between each nanotip and copper substrate. So the graphene layer is prone to be stressed by electrostatic field. In that situation, some induced “doping” appears 7. These doping zones are to be localized only at the very near bases of CNT and are similar to pn junctions, n side being the substrate, and p side the base of CNT. The CNT growing on several interface is a recent topic. However, some par- tial results seems to show a good adhesion with graphene layer. The graphene in itself may present a very different mechanical resistance, its behavior depending on his structure and layer numbers. The first expected results of such a combination, could be the following : 1. thermal coefficient being 10 times those of copper, the capability to raise the melting temperature over the critical one, so Nottingham effect be- comes a cooling mechanism, 2. form factor being extremely big, will be associated with a great β factor, so the effective electric field will be huge, with a low injected power, 3. electrical resistance of CNT is also expected to be much lower than metals, and a resistive graphene layer is desirable, in order to limit saturating currents in emitters, 4.4 nano displacements and spatial alignments between components Several components need to be aligned with a great precision :", + "injected power, 3. electrical resistance of CNT is also expected to be much lower than metals, and a resistive graphene layer is desirable, in order to limit saturating currents in emitters, 4.4 nano displacements and spatial alignments between components Several components need to be aligned with a great precision : 1. FEA cathode and optics -if any-, 2. FEA cathode and/or optics with DLA, 7a description of induced doping in such a Dirac crystal like graphene is of course very rough. The correct treatment could be the description of Brillouin zones, and the derivation of normal modes 15 3. Laser beam with FEA, 4. Laser beam with DLA An interesting setup was described by [32] in their point projection micro- scope, they decided to nano-position only the laser optics. We notice however that they have not ultra precision criteria between the emitting tip and the specimen. Before describing our proposal, which is finally similar to that reference, let’s explore the possibilities of nano-positioning DLA inside chamber, in link with the best industrial performances. We have analyzed the simple case of aligning the FEA cathode with a standard DLA, for instance a double grating [33]. We have taken for DLA, a standard option of double grating, but any new idea should probably result in similar conclusions. Available precision motions, widely used in microscopy, are piezo electrical translators and rotators, and there are possibilities to assemble them by modu- larity. Let’s consider the precedent simple scheme in the figure 11. x y z' \u0001Y δθ δϕ Figure 11: geometry and freedom degrees in case of independent components inside gun If we want to align properly that set, we must manage a priori these degrees of freedom :lateral X, lateral Y, longitudinal Z displacements relatively to the front entrance of the DLA, and the 2 angles θ and φ of the beam relatively to the DLA axis. In fact we remind that we work with flat beam, so displacement X is not so much precision demanding, we can forget it. We can also work with a fixed Z = Z0 so forget also in Z freedom. Finally, there are 3 freedom degrees left, Y, θ and φ. Accounting Breuer analysis and practical fabrications, the DLA entrance is supposed to be some hundred of nanometers, and -if we play with evanescent waves, the precision of localization is approximately 10nm, in a tuning range of 50 to 150nm, then it is the specification for Y. The typical DLA length being LDLA = 25µm in [33], the first maximal deflection angle θ is given by tg θ LDLA < 20nm 25µm ∼10−3, so θmax = 1mrad ∼ 1 20degree, which is announced as inside performance of rotating piezo compo- nents. The sensibility from other angle φ must be investigated further because it has an action on the 3 components of electromagnetic force impinging on electron and may be responsible of shifting phase and lateral acceleration. 16 Let’s see again the Breuer’s work with his notations, equating phase velocity of wave and particle leads (page 19 of [33]) to", + "investigated further because it has an action on the 3 components of electromagnetic force impinging on electron and may be responsible of shifting phase and lateral acceleration. 16 Let’s see again the Breuer’s work with his notations, equating phase velocity of wave and particle leads (page 19 of [33]) to : vph.v v = ω k|| cosφ = βc (1) so k|| = ωcosφ βc = k0cosφ β (2) It is slightly different from Breuer result, because of presence of cosφ on numerator, not denominator. The results are, to that step, identical on only when φ = 0. We go on by defining k⊥= k0 p (1 −cos2φ β2 ) = k0 p (1 −cos4φ ˜β2 ) = k0 ˜β˜γ ∗ p (1 −˜γ2(1 + cos4φ)) (3) where ˜β = βcosφ and ˜γ = (1 −˜β2)−1 2 We stop here and see what it implies on physical insight. We have ˜β < 1 and ˜γ is real, so it is yet pertinent to speak of evanescent waves and define a distance δ such we find for φ = 0. We also note that for φ ∼π 2 , the parallel wave component becomes negligible, and the wave tends to evolve to plane wave, with dispersion k2 ⊥−ω2 c2 ∼k2 0 −ω2 c2 = 0, the consequence being that acceleration is a priori not possible in that configuration. 8 δ ∼ i k⊥ = ˜β˜γ λ 2π (4) Here we still globally agree with the referenced results, defining evanescent regime with the same expression for the distance δ, but only for φ = 0, which is of practical importance. But there are some divergences when we want to compute the electromag- netic fields, specially its sensibility to angle δφ, with φ ̸= 0 in absence of particle. There are several directions to estimate them. 4.4.0.1 geometrical analysis In that analysis, we follow Breuer’s analysis, and with a similar computation, we obtain : E =    c ˜β˜γ p (1 −˜γ2(1 + cos4φ)) ∗By Ey −c∗cosφ ˜β ∗By    (5) B =    − 1 c∗˜β˜γ p (1 −˜γ2(1 + cos4φ)) ∗Ey By + cosφ c∗˜β ∗Ey    (6) Next step is to compute the force F = E + v × B (7) and obtain 8we can also make another change of variable, defining ˜β = cosφ β . The dispersion relation k2 ⊥= k2 0 −k2 || drives us to k⊥= k0 ˜γ , with same algebraic manipulations. However, here γ ∈[−inf, + inf], so existence of direct and evanescent waves is also shown possible 17 F q =   c ∗αz ∗By + sinφ ∗Ey Ey(1 −cosφ) −c2 v ∗By + Ey ∗(1 + βsinφ)   (8) where we have taken k⊥= (αx, αy, αz) = ( cosφ ˜β , 0, 1 ˜β˜γ ∗ p (1−γ2(1+cos4φ))) and αz is purely imaginary number. That result, expressed in wave coordinates -same conclusion applies in par- ticle coordinate- seems to show that : 1. although magnetic components are out of phase for x and", + "k⊥= (αx, αy, αz) = ( cosφ ˜β , 0, 1 ˜β˜γ ∗ p (1−γ2(1+cos4φ))) and αz is purely imaginary number. That result, expressed in wave coordinates -same conclusion applies in par- ticle coordinate- seems to show that : 1. although magnetic components are out of phase for x and z components of the force, electric components are not strictly always out of phase in that case, 2. there is a transverse force associated to accelerating one Letting out of discussion the vertical force, which could be balanced in a double grating, we have to evaluate the ratio of Fy Fx in the particle coordinates. Projecting it, and developing αz in powers of φ for low incidence angles, we find that ratio : αz ∼i ∗sgn(φ) β p (1 + β2) ∗(1 −1 2 ∗(1 + β2) ∗(φ2 + . . .)) (9) and finally Fy Fx ∼tgφ∗Ey(1 −cosφ) c ∗ 1 i ∗sgn(φ) β ∗ p (1 + β2) ∗(1 − 1 2(1+β2) ∗(φ2 + . . .) ∗By + sinφ ∗Ey (10) then finally Fy Fx ∼φ3 ∗−i ∗sgn(φ) ∗ Ey c ∗By ∗ β p (1 + β2 (11) In a nano-positionning scheme inside the cell, we see then that the deflect- ing force is out of phase with accelerating one -what was already guessed - but of high order with φ for low deflection angles. It is a positive result which shows that stability and precision may be reach, but in other side we asked dimensioning to a supplier, and minimum dimensions of an assembling of pre- cision translators/rotators was announced to be at less 10cm, which represent unfortunately the major part of our chamber. It is why we propose a complete integration of photonics -FEA and DLA- by an overall fabrication, and we report the freedom degrees on laser lenses, so there will be a complimentary evaluation to do with that scheme. 4.4.0.2 Some glance for more adapted analysis Inside that approach, we estimate that the (double) grating configuration are subject to the scattering theory with laser wavelength ⪰1µm greater than grating characteristic dimen- sions λp ⪰200nm, in the near field zone, but in neither of the classical near field zones, static, induction one. 9 9Note also that the femtosecond laser pulse and bunch don’t behave like stationary regimes, and wave packet formalism seems necessary. However, we shall stay inside frequency decom- position, assuming an adiabatic system evolution. 18 Let’s consider the same geometry as figure 11 The coordinate center is taken in the center of a grating; The vector potential may be written as [34] : A(x) = µ0ik X lm h1 l (kr)Ylm(θ, φ) ∗ Z J(x’)jl(kr′)Y ∗ lm(θ′, φ′)d3x’ (12) where h1 l , jl, Ylm are respectively spherical Bessel of third kind, spherical Bessel of first kind and spherical harmonics functions [35], J(x’) the elementary current of source x’, observed at x. The ranges of kr is kr > ǫ where ǫ is physically a minimum distance between the beam and the edges of grating. We can already guess that the forces acting", + "Bessel of first kind and spherical harmonics functions [35], J(x’) the elementary current of source x’, observed at x. The ranges of kr is kr > ǫ where ǫ is physically a minimum distance between the beam and the edges of grating. We can already guess that the forces acting on particle are growing at the DLA entrance, to reach a maximum at the center, and decreases at the output. In fact, there are several domains of approximations depending on quantities kr and kr’. At the extremities, kr ≤1 and in a zone from z −∆z to z + ∆z, where we have kr ∼1. Let’s see a typical value of ∆z. If the beam is approximately centered, x- coordinate is symmetrical so θ = π 2 . Consequently, in the expression of solution A = F(r)P(cosθ)Q(φ), we see that all Legendre polynomials of even order are null and those of odd order are constants < 1. As φ ∼0 in figure 11, we have Q(φ) = e±imφ ∼1 −imφ and only the odd terms hl = hodd l = jl + iyl will survive. The evolution of real and imaginary terms shows that jl < j1 where j1(x) = −sinx x2 −cosx x is an oscillatory function with y1(0) = 0 and y1 < 0.5 for x = kr ∼2, but also yl(x) is a growing suite for x < some λ and have the same behavior as jl for big values of x. Typically if we restrict us to m = 3, the frontier is ∆z = kzδ = 5 Finally hl(kr) ∼iyl(kr) (with always x > ǫ for avoiding divergence). Now in the integrand will have also θ′ = π 2 if we suppose a TEM mode with vertical polarization. We also suppose that laser alignment is perfect, so φ′ = 0 In these conditions A(x) ∝ X lodd,m al(1 −imφ) ∗iyl(kr) ∗F(z, r′) ⊗[ Z jl(kr)Y ∗ lm(π 2 , 0)d3x’] (13) where the convolution by the presence function F(z, r′) is symbolically indi- cating which integral to compute depending on the locus of the sources relatively to the particle inside the three domains. Simplifying for Y ∗ lm, A(x) ∝ X l odd,m al(1 −imφ) ∗iyl(kr) ∗ Z jl(kr)d3x’ (14) The d3x’ domain may be divided in two, inside and outside the zone [−∆, ∆], for summing on z, neglecting the immediate entrance and sortance of the elec- tron. Indeed, in that case, the associated phenomena stays identical whatever the angle φ. There is no necessity to calculate here, the fields components, because we can a posteriori directly inject the vector potential in the hamiltonian of interaction, ie in a scheme in which the particle is present. 19 Note : until now, we rely on the evanescent wave acceleration, to imagine laser acceleration in nano structures. Although it is a practical basis for our system, it is not today demonstrated to be the unique way of field acceleration violating Lawson law, other geometries and coupling could to be evaluated, for instance a ribbon 4.5 Electron optics Inside", + "wave acceleration, to imagine laser acceleration in nano structures. Although it is a practical basis for our system, it is not today demonstrated to be the unique way of field acceleration violating Lawson law, other geometries and coupling could to be evaluated, for instance a ribbon 4.5 Electron optics Inside a RF standard gun for instance, with relativistic output, the longitudinal accelerating electric field is predominant against transverse one. Moreover one can show that Lorentz transformation results in reinforcement of El Et in the particle reference coordinates. Conversely for low energy sources at 10keV, we are not benefiting from that situation. How to alleviate it ? Firstly, at very near near distances from the emitting tips, the electric field, specially the longitudinal one, is very intense. For information only, the figure 12 represents two screenshots of the time-space evolution of electric field magnitude diffracted by an individual tip (h = 2µm, apex = 10nm) at t = 95fs and t = 139fs for a gaussian laser pulse of 100fs width. The simulation was driven under OpenEMS and animation results with Paraview. The observer distance is 2µm. We evaluate the field amplification β ∼11. The gap cathode-anode is 3mm. The peak macroscopic electric field is 3.3M V /m with tip, E Schottky field is -without tip- E gap = 36M V /m. The “amplified” laser field is evaluated to 1.28GV/m without tip amplification, and to 14.1GV/m with tip. Figure 12: Efield transient tip Response to laser gaussian 100fs pulse, opposite plane, times 95 (up) and 139fs That field behavior, evaluated by electromagnetic FDTD tool [36], brings us to distinguish the near field zone where the bunch experiences a strong acceler- ating longitudinal forces, from the remaining space inside our -miniature-gun, 20 Figure 13: Coaxial gun version Figure 14: Integrated version where the acceleration is not sufficient to preserve the bunch cohesion. The typ- ical cathode-anode (iris) distance of our the gun being 3 mm in first draft, large in front of near field one, there is then a critical distance generally lower than 1µm beyond that it becomes necessary to focus the beam, before transporting it through anode iris and presenting it at DLA aperture. A standard focusing optic can’t be easily realized inside 3mm typical cubic volume with miniature focusing apparatus, and more generally will be a proto- type in itself, inside the dimensions of our setup. Moreover, the auto induced stray fields at decimeter extensions of a magnetic or electrostatic lens, could be a very crippling problem. Respecting our “game rule”, ie only low cost laser, nanostructures and dielectric arrays,10 we propose to insert a DLA module inside the gun, at the critical distance of FEA. That proposal scheme is represented with two variants, with and without focusing optics. In coaxial version, figure 13 we have designed a first draft of cylindrical profile, which should have some auto-focusing capabilities. We followed some guidelines, not detailed here but it is to refine further thanks to softwares. In integrated version, figure 14 the very short distance between FEA and DLA avoids the focusing", + "version, figure 13 we have designed a first draft of cylindrical profile, which should have some auto-focusing capabilities. We followed some guidelines, not detailed here but it is to refine further thanks to softwares. In integrated version, figure 14 the very short distance between FEA and DLA avoids the focusing need. Moreover, the strip-line disposition allows to choose a proper input impedance, using the synthesis with Hammerstad equations. We may ask wether it is convenient to immerse in dielectric transparent to 10keV electrons and to laser wavelengths, the zone between FEA and DLA, and the inside of DLA. From one side, fabrication may be easier, and focusing -better or not- is questionable. 11 From other side, the occurrence of collisions of the bunch by centers inside the crystal, essentially inside a length of roughly 1 mm, has to be closely studied in order to determine such a feasibility and the induced losses. * : electrical/electromagnetic behavior of that 1D device remains to ex- plore. We know that depending on laser phase, deflecting, focusing or acceler- 10although some millimetric size realizations has been proposed, nevertheless, these setups necessitates focusing electrodes, so don’t obey to our game rule 11In such an entirely integrated device, looking like a semi-conductor, it could be possible to work without vacuum environment 21 ating forces are impinging on the bunch. Moreover, the linear FEA necessitate asymmetric lens so we must apply specific electron optic. * : as precised in [37], (appendix II in 7th edition) , the wave propagation of electron bunch is not driven by a simple eikonal, the direction of trajecto- ries is not orthogonal to equiphase surfaces, but is rather parallel to electron momentum (which is not necessarily orthogonal to these surfaces). So we see up to now that we shall have to represent the bunch dynamics at low energies, by a Lagrangian approach. Appreciation of Lagrangian or Hamiltonian choice -and even hamiltonian dilemma due to representation of excited states- is left to further discussion. Moreover, the landscape is obscured by charge space and many-bodies interaction inside bunch. All these aspects should be integrated in a single model, also left for next work. 4.6 DLA first stage 4.6.1 THz cells In Thz acceleration, the laser frequency is divided to THz range and interact with the bunch inside a cell with mm dimensions. In such system, a 100 fs laser pulse, for instance, is rectified such the electron bunch “sees” an approximate laser accelerating amplitude during its travel inside the cell. It has a great advantage of a strong acceleration distance and the energy gain may be very high for a 10keV bunch, imagining final energies near relativistic range with a single cell. THz cells benefit from existing experiments. However that process presents some drawbacks : 1. Division of laser frequency inside active crystals, so limitation of incident power, and supplementary cost, 2. Large dimensions of that first stage, not very coherent with in-chip phi- losophy 3. Necessity of focusing between photocathode and THz cell, because the dimensions of that cell don’t allow to realize an integrated", + "Division of laser frequency inside active crystals, so limitation of incident power, and supplementary cost, 2. Large dimensions of that first stage, not very coherent with in-chip phi- losophy 3. Necessity of focusing between photocathode and THz cell, because the dimensions of that cell don’t allow to realize an integrated device Regarding the optical rectification, the necessary energy input for a standard ZnTe crystal is in the 1µJ range for a consistent second order non-linear yield, far from the nJ energy of the source we intend to use, and far also from the Yb fiber lasers planned for the in-chip accelerator. To get that conversion, we might use an ND-Yag amplifier but its PrF is limited to 10Hz. In the following, we deal with non-THz cells. However, THz techniques may be useful and are considered inside 4.7 section. 4.6.2 Traveling wave and direct acceleration in near infrared laser range 4.6.2.1 Analysis In a first step, standard double grating is planned for our experiment, and its geometry is naturally adapted to the proposed 1D flat FEA. In all the following, we shall restrict us to traveling devices, so THz cells are not studied here, they constitute a very different landscape because we cannot consider the material tailoring with propagation. Instead focusing 22 us to propagation we must study the cavity aspect and these structure are fundamentally different from that fact. However, let’s discuss here more generally of freedom degrees for structure fabrication, geometry, material choice, . . . of an “ideal” sub-relativistic acceler- ating stage. Before dealing with it, we point out some interesting topics and research directions. The reference [38] about AVM method, illustrate in another point of view, the mutual interaction between the beam and the total induced field inside ac- celerating structure. Although innovating computation method, it reminds us the well-known problem of beam loading for example and has to be kept in memory in any design; furthermore, it must start from a deliberately chosen initial design. In other side, much work has been done for optical guides, specif- ically with ribbons, not directly transposable for the main e- channel, but at less for light couplers [39]. So in any design, the two quantities Fa as in [38], and normalized emittance ǫ must be specified.12 Now, we dispose of a spatial and temporal profile of entrance bunch, trans- ported through a short critical distance, knowing it is generated by a 1D flat FEA array, we have a set of dielectric volumes, of arbitrary indexes. 13 There is no reason to suppose that sectional XY bunch profile 14 is uniform in X, and of course in Y. Nevertheless, the held symmetries are relative to axis OX and OY, so our device is not 2D equivalent but a volume, and may however keep these quadrant symmetries. Its length is determined by constraint on distortion, then output emittance. It may be evaluated by independent simulations, so the main problem is the determination of sectional S(z) structure of our DLA.15 Also, the suite S(i) takes in account the energy variation of the bunch, like", + "keep these quadrant symmetries. Its length is determined by constraint on distortion, then output emittance. It may be evaluated by independent simulations, so the main problem is the determination of sectional S(z) structure of our DLA.15 Also, the suite S(i) takes in account the energy variation of the bunch, like in the case of chirped gratings. The entrance section S(0) is not a rectangle but must be adapted to entrance profile of the bunch. Which profile have we to choose, knowing that the bunch profile XY is not constant from head to tail ? There are two strategies : 1. the entrance S(0) is vacuum and is sized for maximum section of the bunch, 2. S(0) is partially in vacuum or not at all, for its central part, and its near boundary regions are made in dielectric material, offering a multi-layer index n(r), r > r0 The first strategy is widely used, for instance for the double grating. The second one is somewhat new, although already used in resonant devices. The beam is characterized by a sectional distribution of charges and momenta.16 The other important freedom degree is linked to laser injection and polar- ization. Escaping the technology influence, we define a electric state of the laser field at the frontier of DLA, independent of realization of couplers. We make the 12without degrading generality, for fixing ideas, let’s choose for instance, Fa = G 1GV m ∼0.5, where G is acceleration gradient and E maximum field, and ǫ ∼1nrad.m 13we don’t prohibit meta-materials 14X is chosen for 1D axis of FEA, then Y is vertical axis 15Rigourously we must talk about S(z, t) 16we admit to restrict our generality to cartesian coordinates. It is possible that coordinates adapted to equiphase or momentum gradient surfaces of the e- beam be more pertinent 23 hypothesis that notable component of laser field must be present at all steps of accelerating path, the simplest case being an uniform injection in amplitude.17. Restricting the generality, we study the possibilities of transverse and longitu- dinal laser injections, relatively to electron mean axis beam. laser e- interaction may be indirect, by modes response of the material. For short, the limitations and rough hypothesis of our study are the following : 1. laser injection is transverse or longitudinal 2. the DLA device has Ox and Oy quadrantal symmetries, 3. for beam propagation, we are allowed to cut it in slices of volumes S(z)δz, each one having a profile n(x,y,i), n being the complex index (including meta-materials), and z = iδz 4. for laser injection,18 5. figure of merit are final parameters at DLA output, given emittance and Fa The most general scheme, under our precedent constraints, is suggested by the figure 15 z O x y Figure 15: General case of Dielectric Laser Acceleration 17generalization with non constant field is another difficult step 18even that simplification raises problem because the incident equivalent wave front of elec- tron bunch is not a plane, so cutting in cartesian slices is not the best way to deal with propagation. As a justification, we", + "of Dielectric Laser Acceleration 17generalization with non constant field is another difficult step 18even that simplification raises problem because the incident equivalent wave front of elec- tron bunch is not a plane, so cutting in cartesian slices is not the best way to deal with propagation. As a justification, we can say that each slice thickness may be chosen enough tiny to push in second order, the “aberrations” of the electronic wave front, and to dispersion inside optical cycle. 24 * As in standard interactions models, we have to describe the laser field, the material response to bunch transit, taking account the n(x, y, i) suite, and define a cross mechanism between them. 4.6.2.2 Material choice In fact, we have not an infinite freedom in mate- rials choice, but it should ask for a future comparative study. Dielectric accel- eration lays on high field strength and high flux dielectrics, silicon, AlTiO3,...In order to gain supplementary performances and use complex indexes, we may think of meta-material and photonic combinations with these. Regarding meta-materials [40], the ceramics like TiO2 for instance present negative permitivity, low permeability and consequently a negative index, giving them properties of magnetic mirrors in relatively narrow bands near 500GHz. We must keep in memory that an optical pulse of carrier frequency 400THz and of width, say 100 fs, has an equivalent frequency band of 0.35 1 t = 3.5T Hz, far above the range of these meta-materials, so in the reference it is suggested for high frequencies, to use rather polaritonic materials. Indeed, their frequency range may fit the optical laser spectrum, but there is yet no proof they should be adequate for high fluxes and high electron energies, as most publications refer to micro cavities with some eV band gaps. 4.7 Measurements Asked bunches characterizations are numerous, but the minimal set is : 1. mean current (total charge/second) 2. bunch duration or length 3. bunch position and spatial lateral dimensions of the beam, 4. energy spectrum We emphasize that measurements are not integrated part of our “game rule”, which is devoted to the electron source, so at the outside of vacuum cell, we obviously shall find measurement devices. It seems that effort to precisely de- fine diagnostics for that setup are in nearly same amount that those defining the source in itself. A reason why is the inverse conversion from fast bunches to temporal electrical signals. Ideas are either transposition of standard tech- niques for long bunches, or indirect methods by photonics. We can extrapolate one of them, [41], using a decelerating grating, instead of deflecting one, the decelerating grating plays the role of energy high pass filter. That trick permit measurement simplification, all is done in the axis, and the same laser oscil- lator allows accelerating and decelerating phases. However it is necessary to implement two DLA on the same wafer. 4.7.1 Charge of a bunch Mean current will be known by transposition of standard techniques like Fara- day cup. However, in our very low charges context, we guess it may present several issues. We have analyzed the", + "phases. However it is necessary to implement two DLA on the same wafer. 4.7.1 Charge of a bunch Mean current will be known by transposition of standard techniques like Fara- day cup. However, in our very low charges context, we guess it may present several issues. We have analyzed the reference [42] which gives interesting de- sign elements, even applied to ions. Indeed, we retain the cup geometry and 25 Figure 16: miniature Faraday cup Figure 17: Standard device (from Kindall) the necessity to do some simulations; we shall have also to evaluate if cool- ing is or not necessary. Following the reference, we evaluate roughly the total charge and thermal eventualities. If we want to capture the entire beam, with- out extra focusing, the entrance slit of the cup must stay at very tiny distance from DLA output. By analogy with the temporal dispersion, we know that roughly, a 10keV femtosecond electron beam without manipulation, may be en- larged fastly to 1 ps. The spreading distance, accounting for 0.2 c speed, will be d = 1ps∗0.2c = 60µm. It is of course unrealistic, unless to “envelop” the output by the cup. In other side, Faraday cup is claimed to be able to measure pA cur- rents. In our case, the charge estimated at the entrance of DLA is to be 34mA, we calculated that a 1D Array of 1000 tips, each one delivering 34µA, is conve- nient. The charge collected by a 100 femtosecond pulse has been also evaluated to 0.27 fC, so the mean current by pulse will be < i >pulse= 0.2710−15 10010−15 = 2.7mA. Obviously, that current is by far, above lowest limit of that device, but it is to be verified that the low charge itself, is not a limiting factor, because of possible noisy -fast- charging effects. Of course, with MHz PrF, the situation becomes comfortable. That device, although of reasonable dimensions, say 5cm, kills the beam, so either it must be removable or micro positioned in order to move it away from the beam inside the vacuum cell; alternatively we may realize a set of integrated DLA(s) with and without Faraday cups directly printed on chip. For not integrated solution, a - low-cost - micro positioner appears to be useful. Among the issues raised by the implementation of ultra low currents, the reference points out the noise induced in the measurement cables, the vibra- tions of vacuum pump and the residual components from vacuum to air feed through(s). Exploiting the schematics of the reference, and those from other standard realizations of CERN, we propose a setup by the figure 16. Regarding noise sources, we thrust in vacuum team to choose the adapted vacuum pumps and materials, and feed through. We concentrate on electronic components and we suggest using optical fiber for signal line. its powering may be brought by the polarizing ring, and we found performing fedoras for 26 fiber optics up to 10−9T orr. The signal is digitized inside the apparatus and converted to optical one. The stopping material could be low thickness graphite", + "we suggest using optical fiber for signal line. its powering may be brought by the polarizing ring, and we found performing fedoras for 26 fiber optics up to 10−9T orr. The signal is digitized inside the apparatus and converted to optical one. The stopping material could be low thickness graphite - stopping distance is quasi null at 10 keV - partially covering the DLA output, because Coulomb scattering is intense at this point.19 Another solution is to integrate Faraday cup in-chip, and replace sample by loadlock, one of them having integrated Faraday cup, others not. We notice that the Faraday cup may be used for other measurement like energy spectrum, modulating the repelling voltage grid for secondary electrons. Nevertheless that apparatus doesn’t allow recording sub picosecond fast signal. A phosphor screen could be inserted for measurement of transverse dimensions of the beam. The figure 17 recalls the standard product. 4.7.2 Bunch length Thz techniques have already prove their efficiency for picosecond bunches [43], we propose to extend them to femtosecond measurements. Among the main use- ful parameters are the first phonon resonance conditioning the bandwidth, here 3.5 THz for ZnTe- then enough for 100fs measurements as we saw previously. Other crystals like GaAs may present better bandwidth, and it was showed that 50fs lengths are accessible by these measures. For ultra-short bunches from as to 10fs range, [44] propose a streaking camera based on Auger effect. It seems that Titanium target is well fitted to that measurement, but of course that technique is beam killing. 4.7.3 bunch position and lateral dimensions Photonic techniques seems to be adapted to these measurements, if we exploit the proposal of [45]. As showed in that reference, the signal is essentially sensible to one dimension only, so Implementation may be in X and Y if we use two gratings mutually rotated of 90 degrees, first one for X and second one for Y. However, the processed signal is devoted to X,Y position and may also take account of a point like particle, then for example a “center” of a bunch. If the bunch is extended in size, complexity may rise in processing. However that situation is encountered also in standard electromagnetic BPM, so solutions are to be found. That section is deliberately incomplete. Diagnostics is a full and difficult domain in itself and is to be completed before practical steps. 4.8 Summary and discussion on possible and preferred FEA and DLA configurations The precedent sections enlightened several possibilities of integration of the cou- ple FEA/DLA, which are summarized on figures 18, 19 and 20. In all layouts, the “eye” encloses all measurements apparatus, and is outside of vacuum cham- ber, if any. Obviously, Thz cell (solution a) although ideally efficient in energy, leads to cumbersome setup, difficult to tune, and generating a growing emittance due 19without Faraday cup, a photonic focusing lens is then necessary for the majority of other measurement 27 Figure 18: Setup with THz cell Figure 19: Coaxial setup with DLA DLA inside Vacuum chamber 28 to several drift and focusing spaces, at least", + "to tune, and generating a growing emittance due 19without Faraday cup, a photonic focusing lens is then necessary for the majority of other measurement 27 Figure 18: Setup with THz cell Figure 19: Coaxial setup with DLA DLA inside Vacuum chamber 28 to several drift and focusing spaces, at least for femtosecond bunches. It is also necessary to convert the laser frequency to THz, with enough flux for the DLA. The solution b was our first idea, because our electrical source was planned to connect to coaxial termination. We have seen that nano-positioning of the DLA relatively to FEA may be extremely difficult. Even with nano-positioning of optics, it rather leads to a demonstrator, and can’t become a realistic -perhaps industrial- solution. A more refined stage consists logically to the integration by nano-fabrication of FEA and DLA. Nevertheless, the coaxial setup renders uneasy the laser in- jection, is mechanically complex (even possible, we have studied it), and the anode distance can’t be reduced to micronic range, so divergence of the beam must probably be constrained before next DLA stage. Finally, our contribution, solution c, emphasizes the interest to develop the last solution d, with entirely integrated chip, in a PCB like configuration. The PCB like layout with a buried symmetric strip-line for injection, is compatible with ultra fast sub-nanosecond electrical HV pulses, or RF amplifier output, and we are able to tune the cathode impedance. It seems at first glance that it impedance is easily tailored with strip line, easier than with coaxial cables. However, with the geometries that we dispose, the calculated impedance are low, so either we rise then by some trick, or we deliberately design a short circuit stub. In that last case of course, numerous issues appear, even if we master nano-structure fabrications. 29 5 Conclusion Our contribution is a proposal for a future sub-relativistic 10keV range electron source, compatible with Laser acceleration module made in nano-structure, Di- electric Laser Acceleration (DLA). We have described and justified an experi- mental setup, more advanced than initial demonstrators given in literature, our proposal goes directly to validation of the system. Its components are based on a Field Effect Array (FEA) flat beam photo- cathode, functioning in field emission pulsed-DC mode with RF or ultra-fast HV generator, associated in integrated fabrication with a first stage DLA module. We estimate that fabrication with very short distances between FEA and DLA, brought naturally by integrated fabrication, will avoid the need of additional optics. Overall system should hold inside a decimeter size chamber, including new measurements of beam based on photonics. A future improvement should be to integrate the laser couplers to DLA. In order to reach demanding performances, it is necessary to evaluate new candidates as Carbon Nano Tubes (CNT) for the FEA configuration, but their assembling must be done in link with solid state and surface physics experts. We have left out number of theoretical and simulation issues for other works, to list them, not exhaustively : 1. Initial momentum and energy distribution of electron at the immediate interface, resulting from cathode", + "configuration, but their assembling must be done in link with solid state and surface physics experts. We have left out number of theoretical and simulation issues for other works, to list them, not exhaustively : 1. Initial momentum and energy distribution of electron at the immediate interface, resulting from cathode physics and estimated by abinitio models and simulations, 2. Presence of surface states and influence on ulterior bunch dynamics,20 3. estimation of thermal stresses on cathode under high PrF and emitted charges, 4. electron dynamics in low charge regime, with eventual many-body inter- actions, and setting of a general model, 5. optics evaluation with flat beam and nano-structures,. . . For a general model of emittance dilution including many-bodies if necessary, certainly some numerical developments do exist, it should be interesting to com- pare an analytical Lagrangian approach with them. Among next contributions it should be useful to analyze the precise case of sub-femtoC bunches in term of several characteristic length of that pseudo-plasma, and deduce if eventually an hydrodynamic model is -or not- adapted to these dynamic evolution. We also point that much stress could be reported on performances of clean room processes. A draft proposal program for fabrication has been transmitted to our specialized clean room platform (Institute of Fundamental Electronic, IEF) at Orsay. Outside expertise is of course very welcome in that complex topic. 20intuitively, the surface state presence/predominance is associated with very low energy vacuum emitted electron, those which stay in meta-stable state in the “turning points” of the potential well. 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OPTICS LETTERS, Vol. 37, No. 5, March 2012. [46] Xavier Gonze and Al. Abinit : first-principles approach to material and nanosystem properties. [47] Electricité de France. Documentation. [48] F. Hecht. New development in freefem++. Journal of Numerical Mathe- matics, Volume 20, Issue 3-4, December 2012. 35", + "ELECTRON CLOUD EFFECTS: CODES AND SIMULATIONS AT KEK K. Ohmi, KEK, 1-1 Oho, Tsukuba, 305-0801, Japan Abstract Electron cloud effects had been studied at KEK-Photon Factory since 1995. e-p instability had been studied in proton rings since 1965 in BINP, ISR and PSR. Study of electron cloud effects with the present style, which was based on numerical simulations, started at 1995 in positron storage rings. The instability observed in KEK- PF gave a strong impact to B factories, KEKB and PEP- II, which were final stage of their design in those days. History of cure for electron cloud instability overlapped the progress of luminosity performance in KEKB. The studies on electron cloud codes and simulations in KEK are presented. INTRODUCTION Electron cloud instability in positron storage ring had been observed at KEK-PF since start of positron operation in 1988. It had not been identified as an electron cloud effect. KEK-PF had suffered very strong vertical coupled bunch instability. Design of KEKB completed 1994-1995. The positron beam instability had to be solved to complete KEKB Low Energy Ring (LER) design. M. Izawa, Y. Sato and T. Toyomasu had performed many experiments [1] and studied a model to solve it. They showed that a short-range wake gave observed mode spectra. They studied a model, in which electrons trapping by beam under the condition of electron-ion plasma [2]. K. Ohmi had studied a possible model to explain the instability. Photoelectrons, which are supplied continuously (every passage of bunches) from the chamber wall, can induce strong coupled-bunch instability [3]. The simulation method is transferred to SLAC-LBNL team to study in PEP-II LER immediately. The instability was confirmed by a series of experiments at BEPC in IHEP, China [4], which was collaboration of IHEP and KEK. The electron cloud effects have been studied vigorously in the B factories, KEKB and PEP-II, and then LHC in 1990’s. Operation of B factories started in 1998- 1999. The instability has been observed in KEKB-LER. Weak solenoid coils were wound along the whole ring to protect electrons near the beam. The unstable mode of the coupled bunch instability was clearly related to the solenoid status, ON or OFF [5,6]. Corrective electron motion reflected to beam unstable mode. Single bunch instability has been observed in KEKB- LER [7,8] . Generally fast head-tail instability is caused by merge between 0 and -1 synchrotron sideband modes in positron ring. In the single bunch instability, clear positive side band νy+aνs (16ns. Systematic measurement and simulations in KEKB were published in [5,6], respectively. The electron motion in a weak solenoid field (~<50 G) differs from the one in a drift space. Electrons rotate along the chamber wall and do not approach to the beam. The electron motion reflects the coupled bunch instability signal. Figure 8 shows the horizontal mode spectrum for solenoid OFF: that is, in drift space. The left picture was given by measurement. The right picture was given by simulation for electron cloud in drift space. Vertical spectra for measurement and simulation were similar as horizontal ones in Figure 8. The feature of unstable modes is induced by the wake force with the regular property [10] as shown in Figure 3. The spectrum had very good agreement with simulations. Figure 9 shows two cuts of movie for simulated beam-electron motion in drift", + "were similar as horizontal ones in Figure 8. The feature of unstable modes is induced by the wake force with the regular property [10] as shown in Figure 3. The spectrum had very good agreement with simulations. Figure 9 shows two cuts of movie for simulated beam-electron motion in drift space. Coherent motion between bunches and electron flow is seen. There is a peak of unstable modes near m=200 in both measurement and simulation. The frequency is 24 MHz (m+νy) or 100MHz (h-m-νy). Figure 8: Horizontal mode spectrum in KEKB. Left picture is given by measurement with solenoid OFF [5,6]. Right picture is simulated for electron cloud in a drift space. Figure 9: Two cut of movie of the beam electron motion. The white point indicates the beam position pass through the chamber. Figure 10 shows the horizontal mode spectrum for solenoid ON. The left and right pictures are given for measurement and simulation. Vertical spectra for measurement and simulation were similar as Figure 10. The spectrum had very good agreement with simulations. Unstable mode was measured for various Bz as shown in Figure 11. Figure 12 shows two cuts of movie for simulated beam- electron motion in solenoid magnet Bz=10G. Coherent motion between bunches and electron rotation along the chamber is seen. Figure 10: Horizontal mode spectrum in KEKB. Left picture is given by measurement with solenoid ON [5,6]. Right picture is simulated by electron cloud in solenoid field 10G. Figure 11: Unstable mode as a function of the solenoid strength [5]. 100% means Bz=~50G. Figure 12: Two cut of movie of beam-electron motion in solenoid magnet. Electrons moving in the solenoid field with a central force have two frequencies as follows, Figure 13 shows the two types of frequencies. The chamber radius of KEKB is 5 cm. The upper and lower frequencies correspond to the cyclotron motion and rotation along chamber surface. The measurement showed lower solenoid field induced higher frequency mode as shown in Figure 11; m=70, f=(m+νy)f0=11MHz or (h-m-νy)f0=117MHz for 5 G, and m=30, f=7MHz or 121MHz for 10 G. The lower frequency ω- agrees with lower betatron sideband (m+νy)f0. The wake force with the regular property [10] does not induce the lower sideband. The wake force is opposite direction (irregular) for the shift of the previous bunch. Figure 14 shows the wake force due to electron cloud in solenoid field. The irregular property is explained in Figure 15. A bunch shifted vertically induced electrons from a side, left or right depending on the solenoid polarity. The electrons rotate along the vacuum chamber. The wake force has a negative (opposite from the shift) peak after π/2 rotation. Figure 13: Magnetron frequency as a function of solenoid magnet strength Bz [6]. Plots (a) and (b) depicts frequencies ω+ (~ωc, cyclotron mode) and ω- (slow rotational mode), respectively. Figure 14: Wake force due to electron cloud in solenoid field [6]. Figure 15: Irregular property of the wake force. Electrons induced by a shifted beam and their motion in solenoid field. The wake force is opposite to the shift. SINGLE BUNCH", + "cyclotron mode) and ω- (slow rotational mode), respectively. Figure 14: Wake force due to electron cloud in solenoid field [6]. Figure 15: Irregular property of the wake force. Electrons induced by a shifted beam and their motion in solenoid field. The wake force is opposite to the shift. SINGLE BUNCH INSTABILITY DUE TO ELECTRON CLOUD Single bunch fast head-tail instability caused by electron cloud has been observed in KEKB [7,8,9]. When beam current exceeds a threshold value, emittance increases and synchro-beta side band signal has been observed. The simulation of the fast head-tail instability is performed by solving following equations [8] (6) Each potential of beam and electrons are solved using PIC algorithm. Since beam (1mmx0.1mm) is localized at the chamber center, free boundary condition is employed. Electron cloud is initialized every interactions with beam with a flat distribution ~40σxx60σy. The single bunch instability signal has also been observed in PETRA-III and Cesr-TA, recently. Single bunch instability in KEKB-LER A beam size blow-up had been observed since early stage of KEKB operation around ~1999 [7]. The blow-up limited the luminosity performance. Figure 16 shows beam size blow-up and luminosity limitation in 2000- 2001. Solenoid coils were wound 2000-2001 in 50%-70% of the whole drift space. Left picture shows beam size blowup without (green) and with (red) the weak solenoid field. Threshold of the beam size blow-up increases from 400 to 800mA. Left picture shows luminosity as function of current. Luminosity was saturated around 550mA at 2000 December. After winding solenoid additionally, 50% to 70%, luminosity was not saturated by 700mA at 2001 March. The solenoid coil was wound further after 2001, and covered 95% of drift space at around 2005. Figure 16: Beam size blow-up and luminosity limitation in 2000-2001. Synchro-betatron sideband signal, which indicates head-tail instability caused by electron cloud, has been observed [9]. The synchro-beta signal synchronizes with the beam size blow-up: when the beam size blow-up is suppressed by the solenoid, the sideband disappears, vice versa. Figure 17 shows the betatron and synchrotron sideband spectra along the bunch train. Vertical axis is bunch train, head to tail. Betatron signal, which is left white line, shift positively. It is tune shift due to electron cloud. Right white line is a positive synchrotron sideband, whose frequency is νy+aνs, where 1